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Pseudorandom processes
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Pseudorandom processes Tamás F. Mória ,∗, Gábor J. Székelyb,c a
Department of Probability Theory and Statistics, Eötvös Loránd University, Pázmány Péter s. 1/C, H-1117 Budapest, Hungary b National Science Foundation, 2415 Eisenhower Avenue, Alexandria, VA 22314, United States c Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, H-1053 Budapest, Hungary Received 12 September 2018; received in revised form 22 February 2019; accepted 1 March 2019 Available online xxxx
Abstract Pseudorandom numbers turned out to be very widely applicable substitutes of real random number sequences. In this paper we introduce pseudorandom processes that appear to be random but they just imitate certain properties of random/stochastic processes. This paper is only the first step in replacing random coins of random walks by trigonometric functions. More general functions (“coins”) lead to unexpected new challenges, e.g. in case of Haar coins. c 2019 Elsevier B.V. All rights reserved. ⃝ Keywords: Stochastic process; Limit theorem; Fourier series; Weak convergence; Pseudorandom numbers
1. Introduction √ Our starting point is the random walk approximation of Brownian motion B(t) = lim s[nt] / n as n → ∞ where sn is a random walk, that is, sn is the sum of n iid random variables that take the values +1 and −1 with the same probability 1/2. We can call them random coins or real coins. In this paper we replace them by the most classical orthogonal functions like sine and cosine. It turns out√that their partial sums, Sn and Tn , have limit distributions without dividing them by the usual n factor. This led us to a conjecture on more general 1-periodic functions. In the final sections we discuss orthogonal polynomials and also the Haar series as substitutes of random coins. Our “pseudo-coins” are always identically distributed but not independent, not even exchangeable because then they would be conditionally independent, that is conditionally “real coins”. The price we need to pay for replacing “real coins” with “pseudo-coins” is that limit distributions of pseudo-random series might depend on the order of terms. In many cases our “pseudo-coins” are uncorrelated/orthogonal, for example the trigonometric coins below. ∗ Corresponding author.
E-mail addresses: [email protected] (T.F. Móri), [email protected] (G.J. Székely). https://doi.org/10.1016/j.spa.2019.03.004 c 2019 Elsevier B.V. All rights reserved. 0304-4149/⃝
Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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2. Trigonometric coins and processes Let the probability space be the interval [0, 1) with the Lebesgue measure, ω ∈ [0, 1) and let U, V be independent random variables, uniformly distributed on [0, 1). Then we have the results below. Theorem ∑ 1 (Sine Coins). The sine coins, sin(2kπ ω), k = 1, 2, . . . , are identically distributed, Sn := nk=1 sin(2kπ ω) as n → ∞ has a limit distribution without any normalization (!), and this limit distribution is the same as the distribution of cos(πU ) − cos(2π V ) . X := 2 sin(πU ) Corollary 1. and its pdf f X (x) =
The limit distribution of Sn is a Cauchy distribution with scale parameter 1/2 2 . π (1 + 4x 2 )
Theorem 2 (Cosine Coins). The cosine coins, cos(2kπ ω), k = 1, 2, . . . , are identically ∑ distributed and Tn := nk=1 cos(2kπ ω) as n → ∞ has a limit distribution which is the same as the distribution of sin(2π V ) − sin(πU ) . Y := 2 sin(πU ) Corollary 2.
The pdf of the limit distribution of Tn is ⏐ ⏐ ⏐ ⏐ { } 1 ⏐⏐⏐⏐ 2 ⏐ ⏐ / −1, − 12 , 0 . log ⏐1 + ⏐⏐, x ∈ f Y (x) = 2 ⏐ π |2x + 1| x
This function, unlike the Cauchy pdf, is neither bounded, nor unimodal. It is symmetric around − 21 . Remark 1. Trigonometric coins are not exchangeable and thus it can easily happen that a permutation of the coins sin(2kπ ω) is such that the sum of the first n of them, Sn∗ has a weak limit different from Cauchy. For example, it is known that a sufficiently rare subsequence of sin(2kπ ω) behaves as if they were independent [see3,8], and if we insert the remaining terms √ into the √ essentially independent sequence such that their number until the nth term is o( n) then Sn∗ / n tends to a Gaussian distribution because the CLT applies. We do not know the set of all weak accumulation points of Sn∗ /a(n) where a(n) is a suitable normalizing sequence. For the proofs of our theorems we will need the following lemma where {.} denotes the fractional part. Lemma 1. Let t1 , . . . , tr be positive real numbers, linearly independent over the rational numbers Q. Let ϕ1 , . . . , ϕr be real valued functions such that ϕ j (ν) ∼ νt j as ν → ∞, that is, the ratio of the two sides tends to 1. Finally, let U, V be random variables; U being uniformly distributed on [0, 1], (and V having an absolutely continuous distribution. Then the ) limit distribution of the vector U, {ϕ1 (ν) V }, . . . , {ϕr (ν) V } as ν → ∞ is uniform on the r + 1 dimensional unit cube. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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Proof. It is well known, cf. Ch.1, §7 of Billingsley [5], that for the limit distribution we only need to show ( ) lim E e2πi(m 0 U + m 1 ϕ1 (ν)V + · · · + m r ϕr (ν)V ) = 0 (1) ν→∞
for arbitrary integers m 0 , m 1 , . . . , m r provided not all of them are zero. Now, (1) is obviously true if m 1 = · · · = m r = 0 (and m 0 ̸= 0). If m 1 , . . . , m r are not all zero, then decompose eim 0 2πU into the sum of real and imaginary parts, then take the positive and negative parts of both terms. In this way the expectation above is expressed as a sum of four expectations; the first of them is ( ( )+ ) E e2πi(m 1 ϕ1 (ν)V + · · · + m r ϕr (ν)V ) cos(2m 0 πU ) . This can be expressed as a constant multiple of the integral ∫ e2πi(m 1 ϕ1 (ν) + · · · + m r ϕr (ν))V dµ,
(2)
where µ is the ( )+ probability measure with Radon–Nikodym derivative dµ/dP proportional to cos(2π m 0 U ) . In the exponent m 1 ϕ1 (ν) + · · · + m r ϕr (ν) = ν(m 1 t1 + · · · + m r tr ) + o(ν) diverges to −∞ or +∞ because the coefficient of ν is not 0 by the condition imposed on the numbers ti . The distribution of V with respect to µ is still absolutely continuous. Since the characteristic function of an absolutely continuous probability measure disappears at infinity by the Riemann–Lebesgue lemma, we obtain that (2) tends to 0 as ν → ∞. The remaining three integrals can be handled similarly. □ Proofs of Theorems 1, 2, and Corollaries 1, 2. It is well known that (( ) ) cos(π ω) − cos n + 12 2π ω sin((n + 1)π ω) sin(nπ ω) = . Sn (ω) = sin(π ω) sin(π ω) ( {( ) }) Here U (ω) = ω is uniformly distributed on [0, 1), and the limit distribution of U, n + 12 U as n → ∞ is uniform on the unit square [0, 1)×[0, 1). This is implied by Lemma 1 with r = 1, ϕ1 (ν) = ν+ 12 , and V = U . Hence Theorem 1 follows. For Corollary 1 observe that − cos(2π V ) and cos(π V ) have the same distribution, thus the limit distribution of Sn can also be written in the form cos(πU ) + cos(π V ) . 2 sin(πU ) { } { } Since ξ := 12 (U + V ) and η := 21 (U − V ) are also independent and uniformly distributed on [0, 1) we have cos(πU ) + cos(π V ) cos(π (ξ + η)) + cos(π (ξ − η)) = 2 sin(πU ) 2 sin(π(ξ + η)) cos(π ξ ) cos(π η) = sin(π ξ ) cos(π η) + cos(π ξ ) sin(π η) 1 2 ]. = 1[ tan(π ξ ) + tan(π η) 2
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The denominator, being the arithmetic mean of two independent standard Cauchy distributed random variables, is also standard Cauchy and the reciprocal of a Cauchy random variable is Cauchy again. This proves Corollary 1. The sum of cosines can be treated similarly. From the well-known formula ) ) (( sin n + 12 2π ω 1 − Tn (ω) = 2 sin(π ω) 2 it immediately follows that the limit distribution of Tn is equal to the distribution of Y =
sin(2π V ) − sin(πU ) , 2 sin(πU )
where U and V are independent and uniformly distributed on [0, 1). For computing the probability density function of Y let us first consider the pdf of |2Y + 1|. It is the same as that of sin(π V )/ sin(πU ), because | sin(2π V )| and sin(π V ) are identically distributed. The numerator and the denominator are independent and their common pdf is 2 1 ·√ , 0 < x < 1, π 1 − x2 therefore the pdf of |2Y + 1| at x ∈ (0, 1] is ∫ +∞ ∫ 1 t 4 √ h(x) = |t|g(t)g(t x) dt = 2 dt. 2 π 0 (1 − t )(1 − x 2 t 2 ) −∞ g(x) =
Let us substitute s = t 2 . After some easy calculations we get ∫ 1 1 2 √ ds h(x) = 2 π 0 (1 − s)(1 − x 2 s) [ ( √ )]s=1 √ 4 = − 2 log x 1 − s + 1 − x 2 s π x s=0 2 1+x = 2 log . π x 1−x Since 2Y + 1 = sin(2π V )/ sin(πU ) and its reciprocal are identically distributed, h(1/x) = x 2 h(x). Therefore ⏐1 + x ⏐ 2 x +1 2 ⏐ ⏐ h(x) = 2 log = 2 log ⏐ (3) ⏐ π x x −1 π x 1−x for x > 1. Moreover, the distribution of 2Y + 1 is symmetric around the origin, its pdf is equal to 12 h(|x|), x ∈ R. Finally, the pdf of Y is obviously h(|2x + 1|), from which straightforward calculation leads to the formula of Corollary 2. □ Remark 2. If the random variables sin(2kπ ω) were iid, then by L´evy’s equivalence theorem [6, Theorem 9.5.5] the weak convergence of Sn would imply its almost sure convergence. But our Sn , n = 1, 2, . . . sequence is divergent almost surely. The same holds for Tn . Remark 3. More than ten years ago one of our Ph.D. students, Vidyadhar S. Phadke, gave less transparent proofs of these results in his dissertation [17] based on a theorem of R´enyi [19]. In 2014 we proposed Corollary 1 for a college level contest problem in Hungary [10]. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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We are going to characterize the weak limits of the finite dimensional distributions of the processes X n (t) = S⌊nt⌋ and Yn (t) = T⌊nt⌋ , t ≥ 0, as n → ∞. Since Sn (0) = Tn (0) ≡ 0, we can focus on positive values of t. ) } {( First, introduce the process γn (t, ω) = ⌊nt⌋ + 12 2π ω , t > 0, ω ∈ [0, 1). Theorem 3. Let H be a Hamel basis (maximal subset of the real numbers that is linearly independent over Q, the field of rational numbers). Let t1 , . . . , tk be different positive numbers, and consider the unique representation ∑ tj = α j h h, 1 ≤ j ≤ k, α j h ∈ Q. h∈H
Note that the sum can have finitely many nonzero terms only; Put H′ = {h ∈ H : α j h ̸= 0 for some j = 1, . . . , k}. Suppose α j h = p j h /N , where p j h and (thus N is a common denominator ( N > 0 are integers ) ′ for 1 ≤ j ≤ k and h ∈ H ). Then γ (t ), . . . , γ (t ) converges in distribution to the vector n 1 n k ( ) γ (t1 ), . . . , γ (tk ) , where {∑ } γ (t j ) = h∈H′ p j h Vh , 1 ≤ j ≤ k, and Vh , h ∈ H′ are independent and uniformly distributed on the interval [0, 1). For every fixed t the distribution of γ (t) is uniform on [0, 1), and γ ∗ (t) := γ (e−t ), t ∈ R, is (strongly) stationary. Proof. Since (⌊ ∑ (∑ ⌋ 1) ( n )) ( ) n ⌊nt j ⌋ + 12 ω = p jhh + ω= ω, p j h ϕh N 2 N ′ ′ h∈H
h∈H
where ϕh (ν) = νh + O(1), Lemma 1 can be applied. By Kolmogorov’s extension theorem [4, Section 36] there exists a probability distribution + on the product space [0, 1)R with finite dimensional marginals given by the above theorem. This process γ (t) can be considered the “limiting process” of γn (t), t > 0, as n → ∞, where the word “limit” is meant in the sense of weak convergence of finite dimensional distributions. For more details on this kind of definition ( see ) Doob [7] and [9]. ∗ −t (t) = γ e is stationary, that( is, the joint distribution of The transformed process γ ) ) ( ∗ γ (t1 ), . . . , γ ∗ (tk ) coincides with the distribution of the vector γ ∗ (t1 + h), . . . , γ ∗ (tk + h) , −t j −h h >( 0. This follows from the ) ( fact that using the) notation s j = e , c = e , the distributions of γ (s1 ), . . . , γ (sk ) and γ (cs1 ), . . . , γ (csn ) are also identical because the coordinates of s1 , . . . , sk in the Hamel basis H coincide with those of cs1 , . . . , csk in the Hamel basis cH. □ The following lemma will imply important properties of our pseudorandom processes. Lemma 2. (i) The process γ (t) is not continuous stochastically at any time t (thus γ (t) is “more pathological” than typical continuous or c`adl`ag realization processes that are known to be stochastically continuous). (ii) γ (t) is separable. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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Proof. (i) γ (t) cannot be continuous stochastically, because any positive real number t can be approximated with arbitrary precision by numbers s such that t/s is irrational. In this case γ (t) and γ (s) are independent and identically distributed random variables thus γ (s) cannot converge to γ (t) in probability. (ii) It is enough to find an event N with zero probability, and an everywhere dense countable subset S ⊂ R+ (the separant) such that for every open interval I ⊂ R+ on the complement of N we have inf{γ (t) : t ∈ I } = inf{γ (t) : t ∈ I ∩ S}, sup{γ (t) : t ∈ I } = sup{γ (t) : t ∈ I ∩ S}. Now, let S be an everywhere dense countable set the elements of which are linearly independent over Q, the field of rational numbers. Then γ (t), t ∈ S are (stochastically) independent. Indeed, let us fix a Hamel basis H {and ∑ let t1 , . . . }, tk ∈ S be different positive numbers. By definition, γ (t j ) = h∈H′ p j h Vh , where Vh , h ∈ H are iid and uniform on [0, 1); furthermore, p j h are integers, and p j h ̸= 0 if and only the coefficient of h in the representation ∑ tj = α j h h, 1 ≤ j ≤ k, α j h ∈ Q, h∈H
differs from 0. For every j = 1, . . . , k there exists an h j ∈ H such that α j,h i = 0 for i ̸= j but α j,h j ̸= 0. Therefore the joint conditional distribution of γ (t j ), 1 ≤ j ≤ k, given Vh , h ∈ H \ {h 1 , . . . , h k }, is uniform on the k dimensional hypercube, hence the same holds for the unconditional distribution. Consequently, for every diadic interval I ⊂ R+ we have inf{γ (t) : t ∈ I } = 0,
sup{γ (t) : t ∈ I } = 1
(4)
with probability 1. The number of diadic intervals is countable, so there exists an event N with zero probability such that (4) holds simultaneously for all diadic intervals on the complement of N . Every open interval contains diadic intervals, thus on the complement of N we have inf{γ (t) : t ∈ I ∩ S} = 0 = inf{γ (t) : t ∈ I }, sup{γ (t) : t ∈ I ∩ S} = 1 = sup{γ (t) : t ∈ I } for every open interval I , as needed. □ As we have seen before, (( ) ) cos(π ω) − cos ⌊nt j ⌋ + 21 2π ω S⌊nt j ⌋ (ω) = , 2 sin(π ω) and (( ) ) sin ⌊nt j ⌋ + 21 2π ω − sin(π ω) T⌊nt j ⌋ (ω) = , 2 sin(π ω) hence we immediately obtain the following limit result for the processes X n (t) and Yn (t). Corollary 3.
The “limiting processes” of X n (t) = S⌊nt⌋ and Yn (t) = T⌊nt⌋ , is ( ) ( ) cos(πU ) − cos 2π γ (t) sin 2π γ (t) − sin(πU ) X (t) = , Y (t) = , 2 sin(πU ) 2 sin(πU ) where U is independent of the process γ (t) and uniformly distributed on the interval [0, 1). By the previous lemma X (t), Y (t) are separable but not continuous stochastically. □ Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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3. Fourier coins and processes Theorem 4 (Fourier Coins). Let f be a 1-periodic, twice continuously differentiable function, and ∑n let U be uniformly distributed on the interval [0, 1). Suppose E f (U ) = 0. Then Fn = k=1 f (kU ) converges in distribution. Proof. Expand f into Fourier series: f (ω) =
∞ ∑ [ ] a j sin(2 jπ ω) + b j cos(2 jπ ω) , ω ∈ [0, 1); j=1
the constant term is missing because f (U ) has zero mean. It is well known that this Fourier series converges to f uniformly. Moreover, since f belongs to the Sobolev space H 2 = W 2,2 , it follows that ∞ ∑ j 4 (a 2j + b2j ) < ∞, j=1
see Adams and Fournier [2]. Then by H¨older’s inequality we have ∞ ∞ ∑ ∑ (|a j | + |b j |)1/2 = j(|a j | + |b j |)1/2 · j −1 j=1
j=1
≤
[∑ ∞
4
2
j (|a j | + |b j |)
]1/4 [∑ ∞
j=1
j
−4/3
]3/4
< ∞.
(5)
j=1
Consider the partial sums f N (ω) =
N ∑ ( ) a j sin(2 jπ ω) + b j cos(2 jπ ω) , j=1
Fn,N =
n ∑ k=1
ZN =
f N (kU ) =
n ∑ N ∑ ( ) a j sin(2 jkπU ) + b j cos(2 jkπU ) , k=1 j=1
N ∑
(a j X j + b j Y j ),
j=1
where sin(2 jπ η) − sin( jπ ξ ) cos( jπ ξ ) − cos(2 jπ η) , Yj = , (6) 2 sin( jπ ξ ) 2 sin( jπ ξ ) and ξ , η are independent and uniformly distributed on [0, 1). The X ’s and the Y ’s are identically distributed but dependent. ∑ First we show that the infinite series Z ∞ = ∞ j=1 [a j X j + b j Y j ] converges almost surely. For arbitrary random variables ϕ, ψ define Xj =
ϱ(ϕ, ψ) = min{ε ≥ 0 : P(|ϕ − ψ| > ε) ≤ ε}. This is a complete metric that induces convergence in probability. Now, ϱ(Z N , Z N −1 ) = min{ε ≥ 0 : P(|a N X N + b N Y N | > ε) ≤ ε} ( ) ≤ min{ε ≥ 0 : P (|a N | + |b N |)/| sin(N π ξ )| > ε ≤ ε}. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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Since | sin(N π ξ )| and sin(π ξ ) are identically distributed, and P(c > sin(π ξ )) ≤ c, we have ϱ(Z N , Z N −1 ) ≤ (|a N | + |b N |)1/2 . Similar estimation { holds for ϱ(Fn,N , Fn,N −1 ). Now} apply the Borel–Cantelli lemma to the events A N = |Z N − Z N −1 | > (|a N | + |b N |)1/2 . By (5), the infinite sum of P(A N ) is convergent, hence |Z N − Z N −1 | ≤ (|a N | + |b N |)1/2 if N large enough, and (5) implies almost sure convergence. Let L(ϕ, ψ) denote the L´evy distance between the distributions of the random variables ϕ, ψ, i.e., L(ϕ, ψ) = min{ε ≥ 0 : P(ϕ ≤ t − ε) − ε ≤ P(ψ ≤ t) ≤ P(ϕ ≤ t + ε) + ε, ∀t ∈ R}. The L´evy distance is a complete metric that induces the convergence in distribution. Then L(ϕ, ψ) ≤ ϱ(ϕ, ψ), thus L(Fn , Fn,N ) ≤ L(Z N , Z ∞ ) ≤
∞ ∑ j=N +1 ∞ ∑
(|a j | + |b j |)1/2 , (|a j | + |b j |)1/2 .
j=N +1
As we have seen in Theorems 1 and 2, Fn,N =
n ∑ N ∑ [a j sin(2 jkπU ) + b j cos(2 jkπU )] k=1 j=1
=
n N ∑ ∑ [a j sin(2 jkπU ) + b j cos(2 jkπU )] j=1 k=1
) ) (( N [ ∑ cos( jπU ) − cos n + 21 2 jπU = aj 2 sin( jπU ) j=1 ) ) (( ] sin n + 21 2 jπU − sin( jπU ) + bj 2 sin( jπU ) → ZN {( ) } in distribution, because if n + 21 U → V , then also {( ) } { {( ) }} n + 21 jU = j n + 21 U → { j V } in distribution. Therefore [ ] lim sup L(Fn , Z ∞ ) ≤ lim sup L(Fn , Fn,N ) + L(Fn,N , Z N ) + L(Z N , Z ∞ ) n→∞
≤2
n→∞ ∞ ∑
(|a j | + |b j |)1/2 ,
j=N +1
which can be arbitrarily small if N is large enough. □ ∑ [ Remark 4. One might] conjecture that for all 1-periodic L 2 functions f (ω) = ∞ j=1 a j sin ∑n (2 jπω) + b j cos(2 jπω) we have that k=1 f (kω) has a limit distribution as n → ∞, and Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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this limit distribution is the same as the distribution of Z ∞ = Y j are defined in (6).
∑∞
j=1 (a j X j
9
+ b j Y j ), where X j ,
Unfortunately this is not true, shown by the following Theorem. Heuristically, if the nonzero coefficients of a purely sine trigonometric series are sufficiently rare then the terms are essentially independent [see3,8] and thus Z ∞ becomes a weighted sum of (nearly) independent Cauchy variables. This is convergent if and only if the sum of the absolute values of the coefficients is convergent. More precisely, we will prove the following result. Theorem 5. ∞ ( ∑
Let (n j ) be a strictly increasing sequence of positive integers such that ) n j 1/3 <∞ n j+1 j=1 ∑∞ (e.g., n j = ( j!)4 will(do). Then the ) infinite series j=1 (a j X n j +b j Yn j ) converges in distribution ∑∞ if and only if j=1 |a j | + |b j | < ∞, thus the function f (ω) =
∞ ∑ ( ) a j sin(2n j π ω) + b j cos(2n j π ω) j=1
with coefficients in ℓ2 but not in ℓ1 disproves the conjecture. Proof. By [3, Theorem 4.3] there exist iid random variables U j and V j , j = 1, 2, . . . , uniformly distributed on [0, 1), such that ⏐ ⏐ (⏐ ) (⏐ ) P ⏐{n j ξ } − U j ⏐ ≥ 2c j ≤ 2c j , P ⏐{n j η} − V j ⏐ ≥ 2c J ≤ 2c j , j = 1, 2, . . . , where c j = n j /n j+1 . For 0 ≤ u < 1, 0 ≤ v < 1 put sin(2π v) − sin(π u) cos(π u) − cos(2π v) , h 2 (u, v) = . 2 sin(π u) 2 sin(π u) ( ) ( ) = h 1 {n j ξ }, {n j η} and Yn j = h 2 {n j ξ }, {n j η} . We will show that
h 1 (u, v) = Then X n j ∞ ∑
|a j | |X n j − h 1 (U j , V j )| < ∞,
j=1
∞ ∑
|b j | |Yn j − h 2 (U j , V j )| < ∞
(7)
j=1
almost surely. From this it follows that the series ∞ ∞ ∑ ∑ ( ( ) [ )] (a j X n j + b j Yn j ) and a j h 1 {n j ξ }, {n j η} + b j h 2 {n j ξ }, {n j η} j=1
j=1
are equiconvergent in distribution. As the summands of the latter one are independent, the sum converges in distribution if surely. Let us split the second ) ∑and only( if it does almost series into two. The first part is ∞ a h {n ξ }, {n η} ; this is a sum of independent Cauchy j j j=1 j 1 distributed random variables with scale parameters 2a j . Therefore the sum from 1 to N is also Cauchy ∑ with scale parameter 2(|a1 | + · · · + |a N |), thus it converges in distribution if and only if ∞ j=1 |a j | < ∞. ( ) ∑∞ 1 The second part can be written in the form j=1 b j ζ j − 2 , where ζ1 , ζ2 , . . . are symmetrically ∑ distributed variables, and the pdf of |ζ j | is given in (3). Hence the ( iid1 random ) N distribution of j=1 b j ζ j − 2 is symmetric around (b1+· · ·+b N )/2, thus for the convergence Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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∑ in distribution it is necessary that the numerical series ∞ j=1 b j be convergent. To the symmetric ∑∞ part j=1 b j ζ j one can apply the three series theorem. The necessary and sufficient condition for the convergence of this series is the convergence of the numerical series ∞ ∑
P(|b j ζ j | > 1),
j=1
∞ ∑ ( ) E b j ζ j I (|b j ζ j | ≤ 1) ,
∞ ∑
j=1
j=1
( ) Var b j ζ j I (|b j ζ j | ≤ 1) ,
where I ( . ) stands for the indicator of the event in brackets. By (3), 4|b j | G j (x) := P(|b j ζ j | > x) = P(|ζ j | > x/|b j |) ∼ 2 π x ( ) as j → ∞ and x remains fixed. Moreover, E b j ζ j I (|b j ζ j | ≤ 1) = 0 by the symmetry of ζ j . Finally, ∫ 1 ( ) ( ) Var b j ζ j I (|b j ζ j | ≤ 1) = E b2j ζ j2 I (|b j ζ j | ≤ 1) = 2x G j (x) dx = O(|b j |). 0
Therefore j=1 |b j | < ∞ is sufficient and necessary for the convergence of {n j η}). For the proof of (7) apply the mean value theorem. ⏐ ⏐ ( ) ⏐ X n − h 1 (U j , V j )⏐ ≤ sup ∥h ′ ∥ · {n j ξ } − U j , {n j η} − V j , 1 j ∑∞
∑∞
j=1
b j h 2 ({n j ξ },
( ) where the supremum of the derivative h ′1 is taken over the segment connecting {n j ξ }, {n j η} and (U j , V j ). Here ( ) sin(2π v) π 1 − cos(π u) cos(2π v) ′ ,π , h 1 (u, v) = − 2 sin(π u) sin2 (π u) thus ∥h ′1 (u, v)∥ ≤
21/2 π sin2 (π u)
−2/3
≤ 21/2 π c j
,
1/3
provided sin(π u) ≥ c j . Now, let A j denote the event that at least one of the following conditions is violated: ⏐ ⏐ ⏐ 1/3 1/3 ⏐ sin(π {n j ξ }) ≥ c J , sin(πU ) ≥ c J , ⏐{n j ξ } − U j ⏐ < 2c j , ⏐{n j η} − V j ⏐ < 2c j . ∑ 1/3 Then P(A j ) ≤ 2c j + 4c j , hence ∞ j=1 P(A j ) < ∞. By the Borel–Cantelli lemma it follows almost surely that only finitely many of the events A j can occur. On the other hand, on the complement of A j we have ⏐ ⏐ ⏐ X n − h 1 (U j , V j )⏐ ≤ 21/2 π c−2/3 81/2 c j = 4π c1/3 , j j j ∑∞ thus j=1 |a j | |X n j − h 1 (U j , V j )| < ∞ holds almost surely by assumption. ⏐ ⏐ ( 1/3 ) In the same way one can show that ⏐Yn j − h 2 (U j , V j )⏐ = O c j , thus completing the proof. □ Motivated by this counterexample, we conjecture the following. ) ∑∞ ( ∑ Conjecture 1. If f (ω) = a sin(2 jπ ω) + b cos(2 jπ ω) where ∞ j j j=1 j=1 (|a j | + |b j |) < ∞, ∑n then k=1 f (kω) has a limit distribution as n → ∞, and this limit distribution is the same as Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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the distribution of Z ∞ = Xj =
∑∞
j=1 (a j X j
cos( jπ ξ ) − cos(2 jπ η) , 2 sin( jπ ξ )
11
+ b j Y j ), where Yj =
sin(2 jπ η) − sin( jπ ξ ) . 2 sin( jπ ξ )
and ξ , η are independent and uniformly distributed on [0, 1). ∑∞ The condition j=1 (|a j | + |b j |) < ∞ in the conjecture is equivalent to the absolute convergence of the Fourier series. This condition holds, e.g., if f is H¨older continuous with exponent a > 1/2 or if f is of bounded variation and H¨older continuous with exponent a > 0 [11]. (a-H¨older continuity means that | f (x) − f (y)| ≤ C|x − y|a .) The conjecture does not apply e.g. to the square wave function f (ω) = sgn sin(2π ω) because f (ω) =
∞ 4 ∑ sin((2 j − 1)2π ω) , π j=1 2j − 1
and thus the sum of the Fourier coefficients diverges. We do know that an exponentially rare subsequence, the Rademacher system: rn (ω) = sgn sin(2n π ω) is iid thus for a suitable permutation of the terms sgn sin(2nπ ω) the CLT applies. In a recent paper of Pillai and Meng [18] the authors prove that if (A1 , . . . , An ) and (B1 , . . . , Bn ) are iid multivariate normal N (0, Σ ) with positive main diagonal in Σ , and a j ≥ 0, j = 1, 2, . . . , n, then the sum n ∑ j=1
aj
Aj Bj
is Cauchy distributed with scale parameter This led us to the following conjecture.
∑n
j=1
aj.
∑∞ Conjecture 2. If f (ω)∑= ω) is a pure sine Fourier series with a j ≥ 0, j=1 a j sin(2 jπ∑ n j = 1, 2, . . . , and a := ∞ a < ∞, then j j=1 k=1 f (kω) has a limit distribution as n → ∞, and this limit distribution is a/2 times standard Cauchy. Simulations, e.g. on square wave functions, suggest the following ∫1 Conjecture 3. Let f be a 1-periodic L 2 function with 0 f (ω) dω = 0, and let c(n) ∑denote the sum of the absolute values of the first n Fourier coefficients of f . Then (1/c(n)) nk=1 f (kω) has a non-degenerate limit distribution. Remark 5. Kesten (1960,1962) showed that if f (ω) = I (a ≤ ω ≤ b) − (b − a) is a centered and periodically extended indicator function of a subinterval [a, b] ⊂ [0, 1) than the limit distribution of n 1 ∑ f (kπU + V ) log n k=1 is Cauchy whenever U, V are independent and uniformly distributed on [0, 1). According to ∑nKesten: “It seems still impossible to say anything about the asymptotic behavior of k=1 f (kπU + V ) [. . . ] for fixed V and only one random variable U ”. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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Theorem 6 (Fourier Process). Under the conditions of Theorem 4 the finite dimensional distributions of Fn (t) = F⌊nt⌋ weakly converge to those of the process ] ∞ [ ∑ cos( jπU ) − cos(2 jπ γ (t)) sin(2 jπ γ (t)) − sin( jπU ) F(t) = aj + bj 2 sin( jπU ) 2 sin( jπU ) j=1 where γ (t) is defined in Theorem 3, U is uniformly distributed on [0, 1) and it is independent of the process γ (t). Proof. This theorem can be proved by the same method we applied for the proof of Theorem 4. This time Fn , Fn,N and Z N are replaced by vectors, and we have to use the multivariate version of the distances ϱ and L. For random vectors ϕ and ψ let ϱ(ϕ, ψ) = min{ε ≥ 0 : P(max |ϕi − ψi | > ε) ≤ ε}, i
L(ϕ, ψ) = min{ε ≥ 0 : P(ϕi ≤ ti − ε, ∀i) − ε ≤ P(ψi ≤ ti , ∀i) ≤ P(ϕi ≤ ti + ε, ∀i) + ε, ∀t ∈ Rk }, then L(ϕ, ψ) ≤ ϱ(ϕ, ψ) remains valid. □ 4. Chebyshev coins and processes Here we give two examples for orthogonal polynomial coins, Chebyshev polynomials of the first kind and Chebyshev polynomials of the second kind. This time let the probability space be the interval [−1, 1] equipped with the σ -field of Lebesgue measurable sets and half of the Lebesgue measure as probability. Theorem 7 (Chebyshev Coins). Let pn (ω) = cos(n arccos ω), n = 0, 1, 2, . . . be Chebyshev polynomials of the first kind that are orthogonal with respect to the weight function (1−ω2 )−1/2 on [−1, 1] then p1 + · · · + pn has a limit distribution as n → ∞, with pdf 2h 1 (2x + 1), where ⎧ 1 ⎪ ⎪ , if |x| ≥ 1, ⎨ 2|x|3 (8) h 1 (x) = ) √ 1 ( ⎪ ⎪ 2 , if 0 < |x| < 1. 1 − x arcsin x − x ⎩ π |x|3 A similar result is true if we consider Chebyshev polynomials of the second kind, qn , that are orthogonal with respect to the weight function (1 − ω2 )1/2 on [−1, 1], the limit distribution, however, will be different. Proof. Let us start with the summation formula (( ) ) (( ) ) sin n + 21 arccos ω sin n + 12 arccos ω 1 1 (1 ) − = p1 (ω) + · · · + pn (ω) = − . √ 2 2 2(1 − ω) 2 sin 2 arccos ω Consider the fraction in the right-hand side. First we prove that the numerator and the denominator are asymptotically independent. More precisely, (the joint limit√distribution of the ) numerator and the denominator is equal to the distribution of sin(2π V ), 2 U , where U and V are independent and uniformly distributed on [0, 1). This follows from Lemma 1 with r = 1, 1 ϕ(n) = n + 12 , U (ω) = 12 (1 − ω) and V (ω) = 2π arccos(ω). Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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The pdf of sin(2π V ) is 1 1 ·√ , −1 < x < 1, π 1 − x2 √ √ while that of U is 2x, 0 < x < 1. Let h 1 denote the pdf of sin(2π V )/ U . This random variable is symmetrically distributed, thus h 1 (x) = h 1 (−x). Let x > 0 and z = min{1, x}. Then we have ∫ z/x ∫ z 1 y2 2 √ t· √ h 1 (x) = · 2t dt = dt π x3 0 π 1 − t2x2 1 − y2 0 ) √ 1 ( 2 1 − z = arcsin z − z 4π x 3 after some computation. This immediately gives (8). The sum of Chebyshev polynomials of the second kind, qn (ω) =
sin((n + 1) arccos ω) , sin(arccos ω)
can be obtained from the sine summation formula. ) ) ( ) (( cos 21 arccos ω − cos n + 23 arccos ω ( ) q1 (ω) + · · · + qn (ω) = −1 2 sin 21 arccos ω sin(arccos ω) ) ) (( cos n + 23 arccos ω 1 = − − 1. √ 2(1 − ω) (1 − ω) 2(1 + ω) Again, the cosine term in the right-hand side is asymptotically independent of the rest, thus the limit distribution is of the form ) ( 1 cos(2π V ) − 1, 1− √ 4(1 − U ) U where U and V are independent and uniformly distributed on [0, 1). The explicit form of the pdf is left to the reader. □ Remark 6. ∞ ∑
A formal substitution t = 1 into the well-known generating functions
pn (ω)t n =
n=1
1 − tω − 1, 1 − 2tω + t 2
∞ ∑
qn (ω)t n =
n=1
1 −1 1 − 2tω + t 2
(see [1], p. 783) would lead to the (erroneous) deduction that the limit distribution of the sums of Chebyshev polynomials of the first kind is degenerate, concentrated on −1/2, or the limit distribution corresponding to Chebyshev polynomials of the second kind is equal to the 1 distribution of 2U − 1, where U is uniform on [0, 1). Theorem 8 (Chebyshev Process). Define the summation processes of Chebyshev polynomials of first and second kind as Pn (t) =
[nt] ∑ j=1
p j , t > 0,
Q n (t) =
[nt] ∑
q j , t > 0,
j=1
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respectively. Then Pn (t) → P(t) and Q n (t) → Q(t) as n → ∞ in the sense of weak convergence of finite dimensional distributions, where ) ( cos(2π γ (t)) sin(2π γ (t)) 1 1 − 1, − , Q(t) = 1− P(t) = √ √ 2 4(1 − U ) 2 U U γ (t) is defined in Theorem 3, U is uniformly distributed on [0, 1) and it is independent of the process γ (t). Proof. For Chebyshev polynomials of the first kind we can follow the first part of the proof of Theorem 7, applying Theorem 3. In the case of Chebyshev{(polynomials ) of the }second kind, we only have to repeat the proof of Theorem 3 to show that [nt] + 32 arccos ω → γ (t), as n → ∞, and the limit process is asymptotically independent of U (ω) = 21 (1 + ω). Details of the computations are straightforward, therefore we omit them. □ 5. Haar coins Our final example is the Haar orthogonal system whose limiting behavior is very much different from the behaviors in Section 2– 4. Change the probability space back to [0, 1). Put χ0 (ω) ≡ 1. For n > 0 let n = 2k + m, where 0 ≤ m < 2k , and define ⎧ ( ) k/2 ⎪ if m2−k ≤ ω < m + 21 2−k ⎨2 , ( ) χn (ω) = −2−k/2 , if m + 12 2−k ≤ ω < (m + 1)2−k , ⎪ ⎩ 0 otherwise. Clearly, k −1 2∑
χn = 2(k−1)/2rk ,
n=2k−1
where rk stands for the kth Rademacher function defined as rk (ω) = sgn sin(2k π ω). We are going to discuss the asymptotic distribution of the normalized partial sums n 1 ∑ χi , Kn = √ n i=0 √ and will see that K n / n does not have a limit distribution, it is only “merging”, in other words it has many accumulation distributions. None of them are scale mixtures of normals. For a more precise statement introduce
−k/2
Wk = 2
k −1 2∑
n=0
χn = 2−k/2 +
k ∑
2−i/2 rk−i+1 .
i=1
∑ −k/2 Obviously, Wk converges in distribution to the infinite sum W = ∞ rk . One can easily k=1 2 see that ⎧√ ⎨ 2k+1 W on the interval A = [0, (m + 1)2−k ), k+1 K n = √ kn ⎩ 2 W on the interval A = [(m + 1)2−k , 1). n
k
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In a suitably enlarged probability space define the random variables Z (a), 0 ≤ a < 1 by { 2(1−a)/2 with probability 2a − 1, Z (a) = −a/2 2 with probability 2 − 2a , and let Z (a) be independent of W . Finally, let w(ξ, η) denote the Wasserstein distance between the distributions of the integrable random variables ξ and η, that is, w(ξ, η) is the infimum of E|ξ ′ − η′ | over all pairs (ξ ′ , η′ ), where ξ and ξ ′ are identically distributed, just like η and η′ . Theorem 9. ( ) ( ) w K n , Z ({log2 n})W = O n −1/3 , where {.} denotes the fractional part. Hence, the accumulation distributions of the sequence (K n ) coincide with the distributions of Z (a)W , 0 ≤ a < 1. Remark 7. The accumulation distributions √ √cannot be scale mixtures of normal laws, because they are bounded: |W | < 2 − 1, Z (a) ≤ 2. √ Proof of Theorem 9. Approximate the quantity 2−k n K n = Wk + rk+1 I (A) in two steps. Put ℓ = ⌊k/3⌋ and let j be defined by j2−ℓ ≤ (m + 1)2−k < ( j + 1)2−ℓ , finally let B denote the interval [0, j2−ℓ ). With this notation, our first approximation is k−ℓ ∑
2−i/2rk−i+1 +
i=1
∑
2−i/2rℓ+i+1 + rk+1 I (B).
i=k−ℓ+1
It has the same distribution as Z = W + X I (B), where W , X , and B are independent, and P(X = 1) = P(X = −1) = 1/2. The error of approximation is bounded as follows. ∞ k ⏐ ⏐√ ∑ ∑ ⏐ ⏐ −k 2−i/2 + I (A \ B). 2−i/2 + ⏐ 2 n K n − Z ⏐ ≤ 2−k/2 + i=k−ℓ+1
i=k−ℓ+1
( ) ( ) Its expectation is less than 2 + 5 · 2 = O 2−k/3 = O n −1/3 . The second approximation will be Z ′ = W + X I (B ′ ), where B ′ = [0, m2−k ) (and the independence of W , X and B ′ is supposed). Then E|Z − Z ′ | ≤ 2−ℓ , thus we have (√ ) ( ) w 2−k n Sn , Z ′ = O n −1/3 , −ℓ
−(k−ℓ)/2
and therefore √ ) ( ) ( w Sn , 2k n −1 Z ′ = O n −1/3 .
√ Finally, note that P(B ′ ) = 2a − 1 and 2k n −1√= 2−a/2 , where a = {log2 n}; furthermore, the distribution of W + X coincides with that of 2 W . The proof is completed. □
6. Conclusion Pseudorandom numbers turned out to be very effective substitutes of real random numbers e.g. in Monte Carlo method applications, see von Neumann [16] and Knuth [14, Ch.3]. More recently other pseudorandom versions of important random objects, like random graphs, have also been studied, see e.g. Krivelevich and Sudakov [15]. This paper is the first attempt to provide pseudorandom alternatives of random/stochastic processes in the hope of important applications. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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Acknowledgment Part of this research was based on G´abor J. Sz´ekely work supported by the National Science Foundation, while working at the Foundation. Tam´as F. M´ori was supported by the Hungarian National Research, Development and Innovation Office NKFIH – Grant No. K125569. Finally, the authors are indebted to Craig L. Zirbel and Istv´an Berkes for their constructive suggestions and to the editor, Philip Ernst, for his excellent work in handling our manuscript. References [1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, ISBN: 978-0-486-61272-0, 1972. [2] R.A. Adams, J. Fournier, Sobolev Spaces, second ed., in: Pure and Applied Mathematics, vol. 140, Academic Press, Boston, MA, ISBN: 978-0-120-44143-3, 2003. [3] I. Berkes, On the uniform theory of lacunary series, in: C. Elsholtz, P. Grabner (Eds.), Number Theory – Diophantine Problems, Uniform Distribution and Applications, Springer, Cham, 2017, pp. 137–167, http://dx.doi.org/10.1007/978-3-319-55357-3_6. [4] P. Billingsley, Probability and Measure, fourth ed., Wiley, New York, ISBN: 978-1-118-34191-9, 2012. [5] P. Billingsley, Convergence of Probability Measures, second ed., Wiley, New York, ISBN: 978-1-118-62596-5, 2013. [6] K.L. Chung, A Course in Probability Theory, third ed., Academic Press, San Diego, CA, ISBN: 978-0-121-74151-8, 2000. [7] J.L. Doob, Stochastic Processes, Wiley, New York, 1953, ISBN 978-0-471-52369-0. [8] V.F. Gaposhkin, Lacunary series and independent events, Russ. Math. Surv. 21 (6) (1966) 1–82, http: //dx.doi.org/10.1070/RM1966v021n06ABEH001196. [9] R. Getoor, J l doob: foundations of stochastic processes and probabilistic potential theory, Ann. Probab. 37 (5) (2009) 1647–1663, http://dx.doi.org/10.1214/09-AOP465. [10] János Bolyai Math. Soc, Schweitzer contest in higher mathematics 2014, Mat. Lapok (NS) 21 (1) (2015) 77–88, Hungarian. [11] Y. Katznelson, An Introduction to Harmonic Analysis, third ed., Cambridge Mathematical Library, New York, ISBN: 978-0-521-54359-0, 2004. [12] H. Kesten, Uniform distribution mod 1, Ann. of Math. (2) 71 (3) (1960) 445–471, http://dx.doi.org/10.2307/ 1969938. [13] H. Kesten, Uniform distribution mod 1 (ii), Acta Arith. 7 (1962) 355–380, http://dx.doi.org/10.4064/aa-7-4355-380. [14] D. Knuth, The Art of Computer Programming, Vol 2: Seminumerical Algorithms, third ed., Addison-Wesley, Boston, MA, ISBN: 978-0-201-89684-8, 1997. [15] M. Krivelevich, B. Sudakov, Pseudo-random graphs, in: E. Gy˝ori, G.O.H. Katona, L. Lovász, T. Fleiner (Eds.), More Sets, Graphs and Numbers, in: Bolyai Society Mathematical Studies, vol. 15, Springer, Berlin, Heidelberg, 2006, pp. 199–262, http://dx.doi.org/10.1007/978-3-540-32439-3_10. [16] J von Neumann, Various techniques used in connection with random digits, J Res. Nat. Bur. Stand Appl. Math. Ser. 12 (1951) 36–38. [17] V. Phadke, Non-classical convergence results for sums of dependent random variables (Electronic Thesis Or Dissertation), Bowling Green State University, 2008, https://etd.ohiolink.edu/rws_etd/document/get/ bgsu1224514478/inline (Accessed 16 2018). [18] N.S. Pillai, X.L. Meng, An unexpected encounter with Cauchy and Lévy, Ann. Statist. 44 (5) (2016) 2089–2097, http://dx.doi.org/10.1214/15-AOS1407. [19] A. Rényi, On mixing sequences of sets, Acta Math. Acad. Sci. Hungar. 9 (1958) 215–228, http://dx.doi.org/ 10.1007/BF02023873.
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Pseudorandom processes Tamás F. Mória ,∗, Gábor J. Székelyb,c a
Department of Probability Theory and Statistics, Eötvös Loránd University, Pázmány Péter s. 1/C, H-1117 Budapest, Hungary b National Science Foundation, 2415 Eisenhower Avenue, Alexandria, VA 22314, United States c Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, H-1053 Budapest, Hungary Received 12 September 2018; received in revised form 22 February 2019; accepted 1 March 2019 Available online xxxx
Abstract Pseudorandom numbers turned out to be very widely applicable substitutes of real random number sequences. In this paper we introduce pseudorandom processes that appear to be random but they just imitate certain properties of random/stochastic processes. This paper is only the first step in replacing random coins of random walks by trigonometric functions. More general functions (“coins”) lead to unexpected new challenges, e.g. in case of Haar coins. c 2019 Elsevier B.V. All rights reserved. ⃝ Keywords: Stochastic process; Limit theorem; Fourier series; Weak convergence; Pseudorandom numbers
1. Introduction √ Our starting point is the random walk approximation of Brownian motion B(t) = lim s[nt] / n as n → ∞ where sn is a random walk, that is, sn is the sum of n iid random variables that take the values +1 and −1 with the same probability 1/2. We can call them random coins or real coins. In this paper we replace them by the most classical orthogonal functions like sine and cosine. It turns out√that their partial sums, Sn and Tn , have limit distributions without dividing them by the usual n factor. This led us to a conjecture on more general 1-periodic functions. In the final sections we discuss orthogonal polynomials and also the Haar series as substitutes of random coins. Our “pseudo-coins” are always identically distributed but not independent, not even exchangeable because then they would be conditionally independent, that is conditionally “real coins”. The price we need to pay for replacing “real coins” with “pseudo-coins” is that limit distributions of pseudo-random series might depend on the order of terms. In many cases our “pseudo-coins” are uncorrelated/orthogonal, for example the trigonometric coins below. ∗ Corresponding author.
E-mail addresses: [email protected] (T.F. Móri), [email protected] (G.J. Székely). https://doi.org/10.1016/j.spa.2019.03.004 c 2019 Elsevier B.V. All rights reserved. 0304-4149/⃝
Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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2. Trigonometric coins and processes Let the probability space be the interval [0, 1) with the Lebesgue measure, ω ∈ [0, 1) and let U, V be independent random variables, uniformly distributed on [0, 1). Then we have the results below. Theorem ∑ 1 (Sine Coins). The sine coins, sin(2kπ ω), k = 1, 2, . . . , are identically distributed, Sn := nk=1 sin(2kπ ω) as n → ∞ has a limit distribution without any normalization (!), and this limit distribution is the same as the distribution of cos(πU ) − cos(2π V ) . X := 2 sin(πU ) Corollary 1. and its pdf f X (x) =
The limit distribution of Sn is a Cauchy distribution with scale parameter 1/2 2 . π (1 + 4x 2 )
Theorem 2 (Cosine Coins). The cosine coins, cos(2kπ ω), k = 1, 2, . . . , are identically ∑ distributed and Tn := nk=1 cos(2kπ ω) as n → ∞ has a limit distribution which is the same as the distribution of sin(2π V ) − sin(πU ) . Y := 2 sin(πU ) Corollary 2.
The pdf of the limit distribution of Tn is ⏐ ⏐ ⏐ ⏐ { } 1 ⏐⏐⏐⏐ 2 ⏐ ⏐ / −1, − 12 , 0 . log ⏐1 + ⏐⏐, x ∈ f Y (x) = 2 ⏐ π |2x + 1| x
This function, unlike the Cauchy pdf, is neither bounded, nor unimodal. It is symmetric around − 21 . Remark 1. Trigonometric coins are not exchangeable and thus it can easily happen that a permutation of the coins sin(2kπ ω) is such that the sum of the first n of them, Sn∗ has a weak limit different from Cauchy. For example, it is known that a sufficiently rare subsequence of sin(2kπ ω) behaves as if they were independent [see3,8], and if we insert the remaining terms √ into the √ essentially independent sequence such that their number until the nth term is o( n) then Sn∗ / n tends to a Gaussian distribution because the CLT applies. We do not know the set of all weak accumulation points of Sn∗ /a(n) where a(n) is a suitable normalizing sequence. For the proofs of our theorems we will need the following lemma where {.} denotes the fractional part. Lemma 1. Let t1 , . . . , tr be positive real numbers, linearly independent over the rational numbers Q. Let ϕ1 , . . . , ϕr be real valued functions such that ϕ j (ν) ∼ νt j as ν → ∞, that is, the ratio of the two sides tends to 1. Finally, let U, V be random variables; U being uniformly distributed on [0, 1], (and V having an absolutely continuous distribution. Then the ) limit distribution of the vector U, {ϕ1 (ν) V }, . . . , {ϕr (ν) V } as ν → ∞ is uniform on the r + 1 dimensional unit cube. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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3
Proof. It is well known, cf. Ch.1, §7 of Billingsley [5], that for the limit distribution we only need to show ( ) lim E e2πi(m 0 U + m 1 ϕ1 (ν)V + · · · + m r ϕr (ν)V ) = 0 (1) ν→∞
for arbitrary integers m 0 , m 1 , . . . , m r provided not all of them are zero. Now, (1) is obviously true if m 1 = · · · = m r = 0 (and m 0 ̸= 0). If m 1 , . . . , m r are not all zero, then decompose eim 0 2πU into the sum of real and imaginary parts, then take the positive and negative parts of both terms. In this way the expectation above is expressed as a sum of four expectations; the first of them is ( ( )+ ) E e2πi(m 1 ϕ1 (ν)V + · · · + m r ϕr (ν)V ) cos(2m 0 πU ) . This can be expressed as a constant multiple of the integral ∫ e2πi(m 1 ϕ1 (ν) + · · · + m r ϕr (ν))V dµ,
(2)
where µ is the ( )+ probability measure with Radon–Nikodym derivative dµ/dP proportional to cos(2π m 0 U ) . In the exponent m 1 ϕ1 (ν) + · · · + m r ϕr (ν) = ν(m 1 t1 + · · · + m r tr ) + o(ν) diverges to −∞ or +∞ because the coefficient of ν is not 0 by the condition imposed on the numbers ti . The distribution of V with respect to µ is still absolutely continuous. Since the characteristic function of an absolutely continuous probability measure disappears at infinity by the Riemann–Lebesgue lemma, we obtain that (2) tends to 0 as ν → ∞. The remaining three integrals can be handled similarly. □ Proofs of Theorems 1, 2, and Corollaries 1, 2. It is well known that (( ) ) cos(π ω) − cos n + 12 2π ω sin((n + 1)π ω) sin(nπ ω) = . Sn (ω) = sin(π ω) sin(π ω) ( {( ) }) Here U (ω) = ω is uniformly distributed on [0, 1), and the limit distribution of U, n + 12 U as n → ∞ is uniform on the unit square [0, 1)×[0, 1). This is implied by Lemma 1 with r = 1, ϕ1 (ν) = ν+ 12 , and V = U . Hence Theorem 1 follows. For Corollary 1 observe that − cos(2π V ) and cos(π V ) have the same distribution, thus the limit distribution of Sn can also be written in the form cos(πU ) + cos(π V ) . 2 sin(πU ) { } { } Since ξ := 12 (U + V ) and η := 21 (U − V ) are also independent and uniformly distributed on [0, 1) we have cos(πU ) + cos(π V ) cos(π (ξ + η)) + cos(π (ξ − η)) = 2 sin(πU ) 2 sin(π(ξ + η)) cos(π ξ ) cos(π η) = sin(π ξ ) cos(π η) + cos(π ξ ) sin(π η) 1 2 ]. = 1[ tan(π ξ ) + tan(π η) 2
Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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The denominator, being the arithmetic mean of two independent standard Cauchy distributed random variables, is also standard Cauchy and the reciprocal of a Cauchy random variable is Cauchy again. This proves Corollary 1. The sum of cosines can be treated similarly. From the well-known formula ) ) (( sin n + 12 2π ω 1 − Tn (ω) = 2 sin(π ω) 2 it immediately follows that the limit distribution of Tn is equal to the distribution of Y =
sin(2π V ) − sin(πU ) , 2 sin(πU )
where U and V are independent and uniformly distributed on [0, 1). For computing the probability density function of Y let us first consider the pdf of |2Y + 1|. It is the same as that of sin(π V )/ sin(πU ), because | sin(2π V )| and sin(π V ) are identically distributed. The numerator and the denominator are independent and their common pdf is 2 1 ·√ , 0 < x < 1, π 1 − x2 therefore the pdf of |2Y + 1| at x ∈ (0, 1] is ∫ +∞ ∫ 1 t 4 √ h(x) = |t|g(t)g(t x) dt = 2 dt. 2 π 0 (1 − t )(1 − x 2 t 2 ) −∞ g(x) =
Let us substitute s = t 2 . After some easy calculations we get ∫ 1 1 2 √ ds h(x) = 2 π 0 (1 − s)(1 − x 2 s) [ ( √ )]s=1 √ 4 = − 2 log x 1 − s + 1 − x 2 s π x s=0 2 1+x = 2 log . π x 1−x Since 2Y + 1 = sin(2π V )/ sin(πU ) and its reciprocal are identically distributed, h(1/x) = x 2 h(x). Therefore ⏐1 + x ⏐ 2 x +1 2 ⏐ ⏐ h(x) = 2 log = 2 log ⏐ (3) ⏐ π x x −1 π x 1−x for x > 1. Moreover, the distribution of 2Y + 1 is symmetric around the origin, its pdf is equal to 12 h(|x|), x ∈ R. Finally, the pdf of Y is obviously h(|2x + 1|), from which straightforward calculation leads to the formula of Corollary 2. □ Remark 2. If the random variables sin(2kπ ω) were iid, then by L´evy’s equivalence theorem [6, Theorem 9.5.5] the weak convergence of Sn would imply its almost sure convergence. But our Sn , n = 1, 2, . . . sequence is divergent almost surely. The same holds for Tn . Remark 3. More than ten years ago one of our Ph.D. students, Vidyadhar S. Phadke, gave less transparent proofs of these results in his dissertation [17] based on a theorem of R´enyi [19]. In 2014 we proposed Corollary 1 for a college level contest problem in Hungary [10]. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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We are going to characterize the weak limits of the finite dimensional distributions of the processes X n (t) = S⌊nt⌋ and Yn (t) = T⌊nt⌋ , t ≥ 0, as n → ∞. Since Sn (0) = Tn (0) ≡ 0, we can focus on positive values of t. ) } {( First, introduce the process γn (t, ω) = ⌊nt⌋ + 12 2π ω , t > 0, ω ∈ [0, 1). Theorem 3. Let H be a Hamel basis (maximal subset of the real numbers that is linearly independent over Q, the field of rational numbers). Let t1 , . . . , tk be different positive numbers, and consider the unique representation ∑ tj = α j h h, 1 ≤ j ≤ k, α j h ∈ Q. h∈H
Note that the sum can have finitely many nonzero terms only; Put H′ = {h ∈ H : α j h ̸= 0 for some j = 1, . . . , k}. Suppose α j h = p j h /N , where p j h and (thus N is a common denominator ( N > 0 are integers ) ′ for 1 ≤ j ≤ k and h ∈ H ). Then γ (t ), . . . , γ (t ) converges in distribution to the vector n 1 n k ( ) γ (t1 ), . . . , γ (tk ) , where {∑ } γ (t j ) = h∈H′ p j h Vh , 1 ≤ j ≤ k, and Vh , h ∈ H′ are independent and uniformly distributed on the interval [0, 1). For every fixed t the distribution of γ (t) is uniform on [0, 1), and γ ∗ (t) := γ (e−t ), t ∈ R, is (strongly) stationary. Proof. Since (⌊ ∑ (∑ ⌋ 1) ( n )) ( ) n ⌊nt j ⌋ + 12 ω = p jhh + ω= ω, p j h ϕh N 2 N ′ ′ h∈H
h∈H
where ϕh (ν) = νh + O(1), Lemma 1 can be applied. By Kolmogorov’s extension theorem [4, Section 36] there exists a probability distribution + on the product space [0, 1)R with finite dimensional marginals given by the above theorem. This process γ (t) can be considered the “limiting process” of γn (t), t > 0, as n → ∞, where the word “limit” is meant in the sense of weak convergence of finite dimensional distributions. For more details on this kind of definition ( see ) Doob [7] and [9]. ∗ −t (t) = γ e is stationary, that( is, the joint distribution of The transformed process γ ) ) ( ∗ γ (t1 ), . . . , γ ∗ (tk ) coincides with the distribution of the vector γ ∗ (t1 + h), . . . , γ ∗ (tk + h) , −t j −h h >( 0. This follows from the ) ( fact that using the) notation s j = e , c = e , the distributions of γ (s1 ), . . . , γ (sk ) and γ (cs1 ), . . . , γ (csn ) are also identical because the coordinates of s1 , . . . , sk in the Hamel basis H coincide with those of cs1 , . . . , csk in the Hamel basis cH. □ The following lemma will imply important properties of our pseudorandom processes. Lemma 2. (i) The process γ (t) is not continuous stochastically at any time t (thus γ (t) is “more pathological” than typical continuous or c`adl`ag realization processes that are known to be stochastically continuous). (ii) γ (t) is separable. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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Proof. (i) γ (t) cannot be continuous stochastically, because any positive real number t can be approximated with arbitrary precision by numbers s such that t/s is irrational. In this case γ (t) and γ (s) are independent and identically distributed random variables thus γ (s) cannot converge to γ (t) in probability. (ii) It is enough to find an event N with zero probability, and an everywhere dense countable subset S ⊂ R+ (the separant) such that for every open interval I ⊂ R+ on the complement of N we have inf{γ (t) : t ∈ I } = inf{γ (t) : t ∈ I ∩ S}, sup{γ (t) : t ∈ I } = sup{γ (t) : t ∈ I ∩ S}. Now, let S be an everywhere dense countable set the elements of which are linearly independent over Q, the field of rational numbers. Then γ (t), t ∈ S are (stochastically) independent. Indeed, let us fix a Hamel basis H {and ∑ let t1 , . . . }, tk ∈ S be different positive numbers. By definition, γ (t j ) = h∈H′ p j h Vh , where Vh , h ∈ H are iid and uniform on [0, 1); furthermore, p j h are integers, and p j h ̸= 0 if and only the coefficient of h in the representation ∑ tj = α j h h, 1 ≤ j ≤ k, α j h ∈ Q, h∈H
differs from 0. For every j = 1, . . . , k there exists an h j ∈ H such that α j,h i = 0 for i ̸= j but α j,h j ̸= 0. Therefore the joint conditional distribution of γ (t j ), 1 ≤ j ≤ k, given Vh , h ∈ H \ {h 1 , . . . , h k }, is uniform on the k dimensional hypercube, hence the same holds for the unconditional distribution. Consequently, for every diadic interval I ⊂ R+ we have inf{γ (t) : t ∈ I } = 0,
sup{γ (t) : t ∈ I } = 1
(4)
with probability 1. The number of diadic intervals is countable, so there exists an event N with zero probability such that (4) holds simultaneously for all diadic intervals on the complement of N . Every open interval contains diadic intervals, thus on the complement of N we have inf{γ (t) : t ∈ I ∩ S} = 0 = inf{γ (t) : t ∈ I }, sup{γ (t) : t ∈ I ∩ S} = 1 = sup{γ (t) : t ∈ I } for every open interval I , as needed. □ As we have seen before, (( ) ) cos(π ω) − cos ⌊nt j ⌋ + 21 2π ω S⌊nt j ⌋ (ω) = , 2 sin(π ω) and (( ) ) sin ⌊nt j ⌋ + 21 2π ω − sin(π ω) T⌊nt j ⌋ (ω) = , 2 sin(π ω) hence we immediately obtain the following limit result for the processes X n (t) and Yn (t). Corollary 3.
The “limiting processes” of X n (t) = S⌊nt⌋ and Yn (t) = T⌊nt⌋ , is ( ) ( ) cos(πU ) − cos 2π γ (t) sin 2π γ (t) − sin(πU ) X (t) = , Y (t) = , 2 sin(πU ) 2 sin(πU ) where U is independent of the process γ (t) and uniformly distributed on the interval [0, 1). By the previous lemma X (t), Y (t) are separable but not continuous stochastically. □ Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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3. Fourier coins and processes Theorem 4 (Fourier Coins). Let f be a 1-periodic, twice continuously differentiable function, and ∑n let U be uniformly distributed on the interval [0, 1). Suppose E f (U ) = 0. Then Fn = k=1 f (kU ) converges in distribution. Proof. Expand f into Fourier series: f (ω) =
∞ ∑ [ ] a j sin(2 jπ ω) + b j cos(2 jπ ω) , ω ∈ [0, 1); j=1
the constant term is missing because f (U ) has zero mean. It is well known that this Fourier series converges to f uniformly. Moreover, since f belongs to the Sobolev space H 2 = W 2,2 , it follows that ∞ ∑ j 4 (a 2j + b2j ) < ∞, j=1
see Adams and Fournier [2]. Then by H¨older’s inequality we have ∞ ∞ ∑ ∑ (|a j | + |b j |)1/2 = j(|a j | + |b j |)1/2 · j −1 j=1
j=1
≤
[∑ ∞
4
2
j (|a j | + |b j |)
]1/4 [∑ ∞
j=1
j
−4/3
]3/4
< ∞.
(5)
j=1
Consider the partial sums f N (ω) =
N ∑ ( ) a j sin(2 jπ ω) + b j cos(2 jπ ω) , j=1
Fn,N =
n ∑ k=1
ZN =
f N (kU ) =
n ∑ N ∑ ( ) a j sin(2 jkπU ) + b j cos(2 jkπU ) , k=1 j=1
N ∑
(a j X j + b j Y j ),
j=1
where sin(2 jπ η) − sin( jπ ξ ) cos( jπ ξ ) − cos(2 jπ η) , Yj = , (6) 2 sin( jπ ξ ) 2 sin( jπ ξ ) and ξ , η are independent and uniformly distributed on [0, 1). The X ’s and the Y ’s are identically distributed but dependent. ∑ First we show that the infinite series Z ∞ = ∞ j=1 [a j X j + b j Y j ] converges almost surely. For arbitrary random variables ϕ, ψ define Xj =
ϱ(ϕ, ψ) = min{ε ≥ 0 : P(|ϕ − ψ| > ε) ≤ ε}. This is a complete metric that induces convergence in probability. Now, ϱ(Z N , Z N −1 ) = min{ε ≥ 0 : P(|a N X N + b N Y N | > ε) ≤ ε} ( ) ≤ min{ε ≥ 0 : P (|a N | + |b N |)/| sin(N π ξ )| > ε ≤ ε}. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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Since | sin(N π ξ )| and sin(π ξ ) are identically distributed, and P(c > sin(π ξ )) ≤ c, we have ϱ(Z N , Z N −1 ) ≤ (|a N | + |b N |)1/2 . Similar estimation { holds for ϱ(Fn,N , Fn,N −1 ). Now} apply the Borel–Cantelli lemma to the events A N = |Z N − Z N −1 | > (|a N | + |b N |)1/2 . By (5), the infinite sum of P(A N ) is convergent, hence |Z N − Z N −1 | ≤ (|a N | + |b N |)1/2 if N large enough, and (5) implies almost sure convergence. Let L(ϕ, ψ) denote the L´evy distance between the distributions of the random variables ϕ, ψ, i.e., L(ϕ, ψ) = min{ε ≥ 0 : P(ϕ ≤ t − ε) − ε ≤ P(ψ ≤ t) ≤ P(ϕ ≤ t + ε) + ε, ∀t ∈ R}. The L´evy distance is a complete metric that induces the convergence in distribution. Then L(ϕ, ψ) ≤ ϱ(ϕ, ψ), thus L(Fn , Fn,N ) ≤ L(Z N , Z ∞ ) ≤
∞ ∑ j=N +1 ∞ ∑
(|a j | + |b j |)1/2 , (|a j | + |b j |)1/2 .
j=N +1
As we have seen in Theorems 1 and 2, Fn,N =
n ∑ N ∑ [a j sin(2 jkπU ) + b j cos(2 jkπU )] k=1 j=1
=
n N ∑ ∑ [a j sin(2 jkπU ) + b j cos(2 jkπU )] j=1 k=1
) ) (( N [ ∑ cos( jπU ) − cos n + 21 2 jπU = aj 2 sin( jπU ) j=1 ) ) (( ] sin n + 21 2 jπU − sin( jπU ) + bj 2 sin( jπU ) → ZN {( ) } in distribution, because if n + 21 U → V , then also {( ) } { {( ) }} n + 21 jU = j n + 21 U → { j V } in distribution. Therefore [ ] lim sup L(Fn , Z ∞ ) ≤ lim sup L(Fn , Fn,N ) + L(Fn,N , Z N ) + L(Z N , Z ∞ ) n→∞
≤2
n→∞ ∞ ∑
(|a j | + |b j |)1/2 ,
j=N +1
which can be arbitrarily small if N is large enough. □ ∑ [ Remark 4. One might] conjecture that for all 1-periodic L 2 functions f (ω) = ∞ j=1 a j sin ∑n (2 jπω) + b j cos(2 jπω) we have that k=1 f (kω) has a limit distribution as n → ∞, and Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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this limit distribution is the same as the distribution of Z ∞ = Y j are defined in (6).
∑∞
j=1 (a j X j
9
+ b j Y j ), where X j ,
Unfortunately this is not true, shown by the following Theorem. Heuristically, if the nonzero coefficients of a purely sine trigonometric series are sufficiently rare then the terms are essentially independent [see3,8] and thus Z ∞ becomes a weighted sum of (nearly) independent Cauchy variables. This is convergent if and only if the sum of the absolute values of the coefficients is convergent. More precisely, we will prove the following result. Theorem 5. ∞ ( ∑
Let (n j ) be a strictly increasing sequence of positive integers such that ) n j 1/3 <∞ n j+1 j=1 ∑∞ (e.g., n j = ( j!)4 will(do). Then the ) infinite series j=1 (a j X n j +b j Yn j ) converges in distribution ∑∞ if and only if j=1 |a j | + |b j | < ∞, thus the function f (ω) =
∞ ∑ ( ) a j sin(2n j π ω) + b j cos(2n j π ω) j=1
with coefficients in ℓ2 but not in ℓ1 disproves the conjecture. Proof. By [3, Theorem 4.3] there exist iid random variables U j and V j , j = 1, 2, . . . , uniformly distributed on [0, 1), such that ⏐ ⏐ (⏐ ) (⏐ ) P ⏐{n j ξ } − U j ⏐ ≥ 2c j ≤ 2c j , P ⏐{n j η} − V j ⏐ ≥ 2c J ≤ 2c j , j = 1, 2, . . . , where c j = n j /n j+1 . For 0 ≤ u < 1, 0 ≤ v < 1 put sin(2π v) − sin(π u) cos(π u) − cos(2π v) , h 2 (u, v) = . 2 sin(π u) 2 sin(π u) ( ) ( ) = h 1 {n j ξ }, {n j η} and Yn j = h 2 {n j ξ }, {n j η} . We will show that
h 1 (u, v) = Then X n j ∞ ∑
|a j | |X n j − h 1 (U j , V j )| < ∞,
j=1
∞ ∑
|b j | |Yn j − h 2 (U j , V j )| < ∞
(7)
j=1
almost surely. From this it follows that the series ∞ ∞ ∑ ∑ ( ( ) [ )] (a j X n j + b j Yn j ) and a j h 1 {n j ξ }, {n j η} + b j h 2 {n j ξ }, {n j η} j=1
j=1
are equiconvergent in distribution. As the summands of the latter one are independent, the sum converges in distribution if surely. Let us split the second ) ∑and only( if it does almost series into two. The first part is ∞ a h {n ξ }, {n η} ; this is a sum of independent Cauchy j j j=1 j 1 distributed random variables with scale parameters 2a j . Therefore the sum from 1 to N is also Cauchy ∑ with scale parameter 2(|a1 | + · · · + |a N |), thus it converges in distribution if and only if ∞ j=1 |a j | < ∞. ( ) ∑∞ 1 The second part can be written in the form j=1 b j ζ j − 2 , where ζ1 , ζ2 , . . . are symmetrically ∑ distributed variables, and the pdf of |ζ j | is given in (3). Hence the ( iid1 random ) N distribution of j=1 b j ζ j − 2 is symmetric around (b1+· · ·+b N )/2, thus for the convergence Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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∑ in distribution it is necessary that the numerical series ∞ j=1 b j be convergent. To the symmetric ∑∞ part j=1 b j ζ j one can apply the three series theorem. The necessary and sufficient condition for the convergence of this series is the convergence of the numerical series ∞ ∑
P(|b j ζ j | > 1),
j=1
∞ ∑ ( ) E b j ζ j I (|b j ζ j | ≤ 1) ,
∞ ∑
j=1
j=1
( ) Var b j ζ j I (|b j ζ j | ≤ 1) ,
where I ( . ) stands for the indicator of the event in brackets. By (3), 4|b j | G j (x) := P(|b j ζ j | > x) = P(|ζ j | > x/|b j |) ∼ 2 π x ( ) as j → ∞ and x remains fixed. Moreover, E b j ζ j I (|b j ζ j | ≤ 1) = 0 by the symmetry of ζ j . Finally, ∫ 1 ( ) ( ) Var b j ζ j I (|b j ζ j | ≤ 1) = E b2j ζ j2 I (|b j ζ j | ≤ 1) = 2x G j (x) dx = O(|b j |). 0
Therefore j=1 |b j | < ∞ is sufficient and necessary for the convergence of {n j η}). For the proof of (7) apply the mean value theorem. ⏐ ⏐ ( ) ⏐ X n − h 1 (U j , V j )⏐ ≤ sup ∥h ′ ∥ · {n j ξ } − U j , {n j η} − V j , 1 j ∑∞
∑∞
j=1
b j h 2 ({n j ξ },
( ) where the supremum of the derivative h ′1 is taken over the segment connecting {n j ξ }, {n j η} and (U j , V j ). Here ( ) sin(2π v) π 1 − cos(π u) cos(2π v) ′ ,π , h 1 (u, v) = − 2 sin(π u) sin2 (π u) thus ∥h ′1 (u, v)∥ ≤
21/2 π sin2 (π u)
−2/3
≤ 21/2 π c j
,
1/3
provided sin(π u) ≥ c j . Now, let A j denote the event that at least one of the following conditions is violated: ⏐ ⏐ ⏐ 1/3 1/3 ⏐ sin(π {n j ξ }) ≥ c J , sin(πU ) ≥ c J , ⏐{n j ξ } − U j ⏐ < 2c j , ⏐{n j η} − V j ⏐ < 2c j . ∑ 1/3 Then P(A j ) ≤ 2c j + 4c j , hence ∞ j=1 P(A j ) < ∞. By the Borel–Cantelli lemma it follows almost surely that only finitely many of the events A j can occur. On the other hand, on the complement of A j we have ⏐ ⏐ ⏐ X n − h 1 (U j , V j )⏐ ≤ 21/2 π c−2/3 81/2 c j = 4π c1/3 , j j j ∑∞ thus j=1 |a j | |X n j − h 1 (U j , V j )| < ∞ holds almost surely by assumption. ⏐ ⏐ ( 1/3 ) In the same way one can show that ⏐Yn j − h 2 (U j , V j )⏐ = O c j , thus completing the proof. □ Motivated by this counterexample, we conjecture the following. ) ∑∞ ( ∑ Conjecture 1. If f (ω) = a sin(2 jπ ω) + b cos(2 jπ ω) where ∞ j j j=1 j=1 (|a j | + |b j |) < ∞, ∑n then k=1 f (kω) has a limit distribution as n → ∞, and this limit distribution is the same as Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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the distribution of Z ∞ = Xj =
∑∞
j=1 (a j X j
cos( jπ ξ ) − cos(2 jπ η) , 2 sin( jπ ξ )
11
+ b j Y j ), where Yj =
sin(2 jπ η) − sin( jπ ξ ) . 2 sin( jπ ξ )
and ξ , η are independent and uniformly distributed on [0, 1). ∑∞ The condition j=1 (|a j | + |b j |) < ∞ in the conjecture is equivalent to the absolute convergence of the Fourier series. This condition holds, e.g., if f is H¨older continuous with exponent a > 1/2 or if f is of bounded variation and H¨older continuous with exponent a > 0 [11]. (a-H¨older continuity means that | f (x) − f (y)| ≤ C|x − y|a .) The conjecture does not apply e.g. to the square wave function f (ω) = sgn sin(2π ω) because f (ω) =
∞ 4 ∑ sin((2 j − 1)2π ω) , π j=1 2j − 1
and thus the sum of the Fourier coefficients diverges. We do know that an exponentially rare subsequence, the Rademacher system: rn (ω) = sgn sin(2n π ω) is iid thus for a suitable permutation of the terms sgn sin(2nπ ω) the CLT applies. In a recent paper of Pillai and Meng [18] the authors prove that if (A1 , . . . , An ) and (B1 , . . . , Bn ) are iid multivariate normal N (0, Σ ) with positive main diagonal in Σ , and a j ≥ 0, j = 1, 2, . . . , n, then the sum n ∑ j=1
aj
Aj Bj
is Cauchy distributed with scale parameter This led us to the following conjecture.
∑n
j=1
aj.
∑∞ Conjecture 2. If f (ω)∑= ω) is a pure sine Fourier series with a j ≥ 0, j=1 a j sin(2 jπ∑ n j = 1, 2, . . . , and a := ∞ a < ∞, then j j=1 k=1 f (kω) has a limit distribution as n → ∞, and this limit distribution is a/2 times standard Cauchy. Simulations, e.g. on square wave functions, suggest the following ∫1 Conjecture 3. Let f be a 1-periodic L 2 function with 0 f (ω) dω = 0, and let c(n) ∑denote the sum of the absolute values of the first n Fourier coefficients of f . Then (1/c(n)) nk=1 f (kω) has a non-degenerate limit distribution. Remark 5. Kesten (1960,1962) showed that if f (ω) = I (a ≤ ω ≤ b) − (b − a) is a centered and periodically extended indicator function of a subinterval [a, b] ⊂ [0, 1) than the limit distribution of n 1 ∑ f (kπU + V ) log n k=1 is Cauchy whenever U, V are independent and uniformly distributed on [0, 1). According to ∑nKesten: “It seems still impossible to say anything about the asymptotic behavior of k=1 f (kπU + V ) [. . . ] for fixed V and only one random variable U ”. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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Theorem 6 (Fourier Process). Under the conditions of Theorem 4 the finite dimensional distributions of Fn (t) = F⌊nt⌋ weakly converge to those of the process ] ∞ [ ∑ cos( jπU ) − cos(2 jπ γ (t)) sin(2 jπ γ (t)) − sin( jπU ) F(t) = aj + bj 2 sin( jπU ) 2 sin( jπU ) j=1 where γ (t) is defined in Theorem 3, U is uniformly distributed on [0, 1) and it is independent of the process γ (t). Proof. This theorem can be proved by the same method we applied for the proof of Theorem 4. This time Fn , Fn,N and Z N are replaced by vectors, and we have to use the multivariate version of the distances ϱ and L. For random vectors ϕ and ψ let ϱ(ϕ, ψ) = min{ε ≥ 0 : P(max |ϕi − ψi | > ε) ≤ ε}, i
L(ϕ, ψ) = min{ε ≥ 0 : P(ϕi ≤ ti − ε, ∀i) − ε ≤ P(ψi ≤ ti , ∀i) ≤ P(ϕi ≤ ti + ε, ∀i) + ε, ∀t ∈ Rk }, then L(ϕ, ψ) ≤ ϱ(ϕ, ψ) remains valid. □ 4. Chebyshev coins and processes Here we give two examples for orthogonal polynomial coins, Chebyshev polynomials of the first kind and Chebyshev polynomials of the second kind. This time let the probability space be the interval [−1, 1] equipped with the σ -field of Lebesgue measurable sets and half of the Lebesgue measure as probability. Theorem 7 (Chebyshev Coins). Let pn (ω) = cos(n arccos ω), n = 0, 1, 2, . . . be Chebyshev polynomials of the first kind that are orthogonal with respect to the weight function (1−ω2 )−1/2 on [−1, 1] then p1 + · · · + pn has a limit distribution as n → ∞, with pdf 2h 1 (2x + 1), where ⎧ 1 ⎪ ⎪ , if |x| ≥ 1, ⎨ 2|x|3 (8) h 1 (x) = ) √ 1 ( ⎪ ⎪ 2 , if 0 < |x| < 1. 1 − x arcsin x − x ⎩ π |x|3 A similar result is true if we consider Chebyshev polynomials of the second kind, qn , that are orthogonal with respect to the weight function (1 − ω2 )1/2 on [−1, 1], the limit distribution, however, will be different. Proof. Let us start with the summation formula (( ) ) (( ) ) sin n + 21 arccos ω sin n + 12 arccos ω 1 1 (1 ) − = p1 (ω) + · · · + pn (ω) = − . √ 2 2 2(1 − ω) 2 sin 2 arccos ω Consider the fraction in the right-hand side. First we prove that the numerator and the denominator are asymptotically independent. More precisely, (the joint limit√distribution of the ) numerator and the denominator is equal to the distribution of sin(2π V ), 2 U , where U and V are independent and uniformly distributed on [0, 1). This follows from Lemma 1 with r = 1, 1 ϕ(n) = n + 12 , U (ω) = 12 (1 − ω) and V (ω) = 2π arccos(ω). Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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The pdf of sin(2π V ) is 1 1 ·√ , −1 < x < 1, π 1 − x2 √ √ while that of U is 2x, 0 < x < 1. Let h 1 denote the pdf of sin(2π V )/ U . This random variable is symmetrically distributed, thus h 1 (x) = h 1 (−x). Let x > 0 and z = min{1, x}. Then we have ∫ z/x ∫ z 1 y2 2 √ t· √ h 1 (x) = · 2t dt = dt π x3 0 π 1 − t2x2 1 − y2 0 ) √ 1 ( 2 1 − z = arcsin z − z 4π x 3 after some computation. This immediately gives (8). The sum of Chebyshev polynomials of the second kind, qn (ω) =
sin((n + 1) arccos ω) , sin(arccos ω)
can be obtained from the sine summation formula. ) ) ( ) (( cos 21 arccos ω − cos n + 23 arccos ω ( ) q1 (ω) + · · · + qn (ω) = −1 2 sin 21 arccos ω sin(arccos ω) ) ) (( cos n + 23 arccos ω 1 = − − 1. √ 2(1 − ω) (1 − ω) 2(1 + ω) Again, the cosine term in the right-hand side is asymptotically independent of the rest, thus the limit distribution is of the form ) ( 1 cos(2π V ) − 1, 1− √ 4(1 − U ) U where U and V are independent and uniformly distributed on [0, 1). The explicit form of the pdf is left to the reader. □ Remark 6. ∞ ∑
A formal substitution t = 1 into the well-known generating functions
pn (ω)t n =
n=1
1 − tω − 1, 1 − 2tω + t 2
∞ ∑
qn (ω)t n =
n=1
1 −1 1 − 2tω + t 2
(see [1], p. 783) would lead to the (erroneous) deduction that the limit distribution of the sums of Chebyshev polynomials of the first kind is degenerate, concentrated on −1/2, or the limit distribution corresponding to Chebyshev polynomials of the second kind is equal to the 1 distribution of 2U − 1, where U is uniform on [0, 1). Theorem 8 (Chebyshev Process). Define the summation processes of Chebyshev polynomials of first and second kind as Pn (t) =
[nt] ∑ j=1
p j , t > 0,
Q n (t) =
[nt] ∑
q j , t > 0,
j=1
Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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respectively. Then Pn (t) → P(t) and Q n (t) → Q(t) as n → ∞ in the sense of weak convergence of finite dimensional distributions, where ) ( cos(2π γ (t)) sin(2π γ (t)) 1 1 − 1, − , Q(t) = 1− P(t) = √ √ 2 4(1 − U ) 2 U U γ (t) is defined in Theorem 3, U is uniformly distributed on [0, 1) and it is independent of the process γ (t). Proof. For Chebyshev polynomials of the first kind we can follow the first part of the proof of Theorem 7, applying Theorem 3. In the case of Chebyshev{(polynomials ) of the }second kind, we only have to repeat the proof of Theorem 3 to show that [nt] + 32 arccos ω → γ (t), as n → ∞, and the limit process is asymptotically independent of U (ω) = 21 (1 + ω). Details of the computations are straightforward, therefore we omit them. □ 5. Haar coins Our final example is the Haar orthogonal system whose limiting behavior is very much different from the behaviors in Section 2– 4. Change the probability space back to [0, 1). Put χ0 (ω) ≡ 1. For n > 0 let n = 2k + m, where 0 ≤ m < 2k , and define ⎧ ( ) k/2 ⎪ if m2−k ≤ ω < m + 21 2−k ⎨2 , ( ) χn (ω) = −2−k/2 , if m + 12 2−k ≤ ω < (m + 1)2−k , ⎪ ⎩ 0 otherwise. Clearly, k −1 2∑
χn = 2(k−1)/2rk ,
n=2k−1
where rk stands for the kth Rademacher function defined as rk (ω) = sgn sin(2k π ω). We are going to discuss the asymptotic distribution of the normalized partial sums n 1 ∑ χi , Kn = √ n i=0 √ and will see that K n / n does not have a limit distribution, it is only “merging”, in other words it has many accumulation distributions. None of them are scale mixtures of normals. For a more precise statement introduce
−k/2
Wk = 2
k −1 2∑
n=0
χn = 2−k/2 +
k ∑
2−i/2 rk−i+1 .
i=1
∑ −k/2 Obviously, Wk converges in distribution to the infinite sum W = ∞ rk . One can easily k=1 2 see that ⎧√ ⎨ 2k+1 W on the interval A = [0, (m + 1)2−k ), k+1 K n = √ kn ⎩ 2 W on the interval A = [(m + 1)2−k , 1). n
k
Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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In a suitably enlarged probability space define the random variables Z (a), 0 ≤ a < 1 by { 2(1−a)/2 with probability 2a − 1, Z (a) = −a/2 2 with probability 2 − 2a , and let Z (a) be independent of W . Finally, let w(ξ, η) denote the Wasserstein distance between the distributions of the integrable random variables ξ and η, that is, w(ξ, η) is the infimum of E|ξ ′ − η′ | over all pairs (ξ ′ , η′ ), where ξ and ξ ′ are identically distributed, just like η and η′ . Theorem 9. ( ) ( ) w K n , Z ({log2 n})W = O n −1/3 , where {.} denotes the fractional part. Hence, the accumulation distributions of the sequence (K n ) coincide with the distributions of Z (a)W , 0 ≤ a < 1. Remark 7. The accumulation distributions √ √cannot be scale mixtures of normal laws, because they are bounded: |W | < 2 − 1, Z (a) ≤ 2. √ Proof of Theorem 9. Approximate the quantity 2−k n K n = Wk + rk+1 I (A) in two steps. Put ℓ = ⌊k/3⌋ and let j be defined by j2−ℓ ≤ (m + 1)2−k < ( j + 1)2−ℓ , finally let B denote the interval [0, j2−ℓ ). With this notation, our first approximation is k−ℓ ∑
2−i/2rk−i+1 +
i=1
∑
2−i/2rℓ+i+1 + rk+1 I (B).
i=k−ℓ+1
It has the same distribution as Z = W + X I (B), where W , X , and B are independent, and P(X = 1) = P(X = −1) = 1/2. The error of approximation is bounded as follows. ∞ k ⏐ ⏐√ ∑ ∑ ⏐ ⏐ −k 2−i/2 + I (A \ B). 2−i/2 + ⏐ 2 n K n − Z ⏐ ≤ 2−k/2 + i=k−ℓ+1
i=k−ℓ+1
( ) ( ) Its expectation is less than 2 + 5 · 2 = O 2−k/3 = O n −1/3 . The second approximation will be Z ′ = W + X I (B ′ ), where B ′ = [0, m2−k ) (and the independence of W , X and B ′ is supposed). Then E|Z − Z ′ | ≤ 2−ℓ , thus we have (√ ) ( ) w 2−k n Sn , Z ′ = O n −1/3 , −ℓ
−(k−ℓ)/2
and therefore √ ) ( ) ( w Sn , 2k n −1 Z ′ = O n −1/3 .
√ Finally, note that P(B ′ ) = 2a − 1 and 2k n −1√= 2−a/2 , where a = {log2 n}; furthermore, the distribution of W + X coincides with that of 2 W . The proof is completed. □
6. Conclusion Pseudorandom numbers turned out to be very effective substitutes of real random numbers e.g. in Monte Carlo method applications, see von Neumann [16] and Knuth [14, Ch.3]. More recently other pseudorandom versions of important random objects, like random graphs, have also been studied, see e.g. Krivelevich and Sudakov [15]. This paper is the first attempt to provide pseudorandom alternatives of random/stochastic processes in the hope of important applications. Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.
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Acknowledgment Part of this research was based on G´abor J. Sz´ekely work supported by the National Science Foundation, while working at the Foundation. Tam´as F. M´ori was supported by the Hungarian National Research, Development and Innovation Office NKFIH – Grant No. K125569. Finally, the authors are indebted to Craig L. Zirbel and Istv´an Berkes for their constructive suggestions and to the editor, Philip Ernst, for his excellent work in handling our manuscript. References [1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, ISBN: 978-0-486-61272-0, 1972. [2] R.A. Adams, J. Fournier, Sobolev Spaces, second ed., in: Pure and Applied Mathematics, vol. 140, Academic Press, Boston, MA, ISBN: 978-0-120-44143-3, 2003. [3] I. Berkes, On the uniform theory of lacunary series, in: C. Elsholtz, P. Grabner (Eds.), Number Theory – Diophantine Problems, Uniform Distribution and Applications, Springer, Cham, 2017, pp. 137–167, http://dx.doi.org/10.1007/978-3-319-55357-3_6. [4] P. Billingsley, Probability and Measure, fourth ed., Wiley, New York, ISBN: 978-1-118-34191-9, 2012. [5] P. Billingsley, Convergence of Probability Measures, second ed., Wiley, New York, ISBN: 978-1-118-62596-5, 2013. [6] K.L. Chung, A Course in Probability Theory, third ed., Academic Press, San Diego, CA, ISBN: 978-0-121-74151-8, 2000. [7] J.L. Doob, Stochastic Processes, Wiley, New York, 1953, ISBN 978-0-471-52369-0. [8] V.F. Gaposhkin, Lacunary series and independent events, Russ. Math. Surv. 21 (6) (1966) 1–82, http: //dx.doi.org/10.1070/RM1966v021n06ABEH001196. [9] R. Getoor, J l doob: foundations of stochastic processes and probabilistic potential theory, Ann. Probab. 37 (5) (2009) 1647–1663, http://dx.doi.org/10.1214/09-AOP465. [10] János Bolyai Math. Soc, Schweitzer contest in higher mathematics 2014, Mat. Lapok (NS) 21 (1) (2015) 77–88, Hungarian. [11] Y. Katznelson, An Introduction to Harmonic Analysis, third ed., Cambridge Mathematical Library, New York, ISBN: 978-0-521-54359-0, 2004. [12] H. Kesten, Uniform distribution mod 1, Ann. of Math. (2) 71 (3) (1960) 445–471, http://dx.doi.org/10.2307/ 1969938. [13] H. Kesten, Uniform distribution mod 1 (ii), Acta Arith. 7 (1962) 355–380, http://dx.doi.org/10.4064/aa-7-4355-380. [14] D. Knuth, The Art of Computer Programming, Vol 2: Seminumerical Algorithms, third ed., Addison-Wesley, Boston, MA, ISBN: 978-0-201-89684-8, 1997. [15] M. Krivelevich, B. Sudakov, Pseudo-random graphs, in: E. Gy˝ori, G.O.H. Katona, L. Lovász, T. Fleiner (Eds.), More Sets, Graphs and Numbers, in: Bolyai Society Mathematical Studies, vol. 15, Springer, Berlin, Heidelberg, 2006, pp. 199–262, http://dx.doi.org/10.1007/978-3-540-32439-3_10. [16] J von Neumann, Various techniques used in connection with random digits, J Res. Nat. Bur. Stand Appl. Math. Ser. 12 (1951) 36–38. [17] V. Phadke, Non-classical convergence results for sums of dependent random variables (Electronic Thesis Or Dissertation), Bowling Green State University, 2008, https://etd.ohiolink.edu/rws_etd/document/get/ bgsu1224514478/inline (Accessed 16 2018). [18] N.S. Pillai, X.L. Meng, An unexpected encounter with Cauchy and Lévy, Ann. Statist. 44 (5) (2016) 2089–2097, http://dx.doi.org/10.1214/15-AOS1407. [19] A. Rényi, On mixing sequences of sets, Acta Math. Acad. Sci. Hungar. 9 (1958) 215–228, http://dx.doi.org/ 10.1007/BF02023873.
Please cite this article as: T.F. M´ori and G.J. Sz´ekely, Pseudorandom processes, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.03.004.