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Large and moderate deviations for modified Engel continued fractions
Statistics and Probability Letters 98 (2015) 98–106
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Large and moderate deviations for modified Engel continued fractions Lulu Fang Department of Mathematics, South China University of Technology, Guangzhou 510640, PR China
article
abstract
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Article history: Received 28 November 2014 Received in revised form 9 December 2014 Accepted 11 December 2014 Available online 24 December 2014
In this paper, we consider the large and moderate deviation principles for modified Engel continued fractions which is a representation of real numbers in number theory. © 2014 Elsevier B.V. All rights reserved.
MSC: 60F10 11A67 11K55 Keywords: Large deviations Moderate deviations Modified Engel continued fractions Good rate function
1. Introduction Hartono et al. (2002) introduced a new continued fraction algorithm with non-decreasing partial quotients, called the Engel continued fraction (ECF for short) expansion. The name of this new continued fraction expansion is originated from the Engel series expansion considered by Erdős et al. (1958), which is a classical representation of real numbers in number theory. Similarly, we can obtain another new continued fractions named the modified ECF expansion by borrowing from the modified Engel series expansion which was studied by Rényi (1962). Here we give the algorithm of the generation of modified ECF expansion. Let T : (0, 1] −→ (0, 1] be the modified ECF map, which is given by 1 Tx = 1 x
+1
·
1 x
1
−
x
,
where [x] denotes the greatest integer not exceeding x. For any real number x ∈ (0, 1] and all n ≥ 1, let d1 (x) = [1/x] and dn+1 (x) = d1 (T n x), where T n denotes the nth iterate of T . Then for every x ∈ (0, 1], the modified ECF map T can generate a continued fraction expansion in the form of x= d1 +
1 d1 + 1
,
. dn−1 + 1 d2 + . . + . dn + . .
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2014.12.015 0167-7152/© 2014 Elsevier B.V. All rights reserved.
(1.1)
L. Fang / Statistics and Probability Letters 98 (2015) 98–106
99
where dn ≥ 1 are positive integers and dn+1 ≥ dn + 1 for all n ≥ 1. The representation (1.1) is said to be the modified ECF expansion of x and dn (x), n ≥ 1 are called the digits of the modified ECF expansion of x. Sometimes we write the form (1.1) as x = ((d1 , d2 , . . . , dn , . . .)). In fact, this modified ECF expansion is a special case of the Oppenheim continued fractions which was studied by Fan et al. (2007). Moreover, some arithmetic and statistical properties of the modified ECF expansion, such as the representation of rational numbers, law of large numbers and central limit theorem were also discussed by Fan et al. (2007). Now we turn to introducing the large and moderate deviation principles. Let {Xn , n ≥ 1} be a sequence of the realvalued random variables defined on the probability space (Ω , F , P). A function I : R → [0, ∞] is called a good rate function if it is lower semicontinuous and has compact level sets. Let {λn , n ≥ 1} be a positive real number sequence with limn→∞ λn = ∞. We say that the sequence {Xn , n ≥ 1} satisfies a large deviation principle (LDP for short) with speed λn and good rate function I under P, if for any Borel set Γ , we have
− inf◦ I (x) ≤ lim inf x∈Γ
n→∞
1
λn
log P(Xn ∈ Γ ) ≤ lim sup n→∞
1
λn
log P(Xn ∈ Γ ) ≤ − sup I (x),
(1.2)
x∈Γ
where Γ ◦ and Γ denotes the interior and the closure of Γ respectively. Formally there is no distinction between the large deviation principle and the moderate deviation principle (MDP for short). Usually LDP characterizes the convergence speed of the law of large numbers, while MDP describes the speed of convergence speed between the law of large numbers and the central limit theorem. We refer the reader to Dembo and Zeitouni (1998), Touchette (2009) and Varadhan (1984) for general backgrounds of large and moderate deviations. In this paper, we denote by (Ω , F , P) a probability space, where Ω = (0, 1], F is the Borel σ -algebra on Ω and P denotes the Lebesgue measure on (Ω , F ). Fan et al. (2007) showed a strong law of large numbers for the digits sequence {dn , n ≥ 1} occurring in the modified ECF expansion, i.e., for almost all x ∈ (0, 1], we have lim
n→∞
1
log dn (x) = 1.
n
And also proved that the central limit theorem holds for the digits sequence {dn , n ≥ 1}. That is, for every y ∈ R, we have
lim P
log dn − n
√
n→∞
1 ≤y = √
2π
n
y
2 e−t /2 dt .
−∞
A natural question is to consider the probability of the event that leads to the study of large deviations for the modified ECF expansion.
log dn n
deviates away from its ergodic mean of 1? This
Theorem 1.1. Let {dn , n ≥ 1} be the digits sequence of the modified ECF expansion. Then speed n and good rate function I (x) =
x − log(x + 1)
+∞
log dn −n n
, n ≥ 1 satisfies a LDP with
if x > −1 otherwise,
(1.3)
under P. As a complement of Theorem 1.1, we give the following MDP result for the modified ECF expansion. Theorem 1.2. Let {dn , n ≥ 1} be the digits sequence of the modified ECF expansion and {an , n ≥ 1} be a positive sequence satisfying an → ∞, Then
log dn −n an
√
an
n log n
→ ∞ and
an n
→ 0.
(1.4)
, n ≥ 1 satisfies an MDP with speed n−1 a2n and good rate function I (x) = x2 /2 under P. √
Remark 1. We may obtain the result if the second condition of an in (1.4) is replaced by an / n → ∞ as n → ∞. However, n we need the condition √nalog → ∞ to avoid the technical difficulties. The main reason is that we cannot find the finer n estimates when we make use of the Gärtner–Ellis Theorem (see Dembo and Zeitouni, 1998, p. 44, Theorem 2.3.6). As an application of Theorem 1.2, we obtain the following corollary. Corollary 1. Let {dn , n ≥ 1} be the digits sequence of the modified ECF expansion and an = np with p ∈ ( 12 , 1). Then
log dn −n an
, n ≥ 1 satisfies an MDP with speed n−1 a2n and good rate function I (x) = x2 /2 under P.
Recently, some authors began to consider the large and moderate deviations related to number theory. For example, Mehrdad and Zhu (2013) established the large and moderate deviations for Erdős–Kac theorem, which is a celebrated result
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about the number of distinct prime factors of a uniformly chosen random integer in number theory. Zhu (2014) and Hu (2015) studied respectively the large deviations and moderate deviations for Engel, Sylvester and Cantor product series, which are the classical representations of real numbers in number theory. Note that the digits sequence occurring in the Engel, Sylvester and Cantor product series forms a Markov chain and the proofs of Zhu (2014) and Hu (2015) rely on this property, while the digits sequence of modified ECF expansion is not a Markov chain. We overcome this difficulty by using a weak property (i.e. Proposition 2.4) in Section 2. 2. Preliminary In this section, we recall some definitions and some arithmetic and metric properties of the modified ECF expansion. We denote (Ω , F , P) as a probability space and use the notation E(ξ ) to denote the expectation of the random variable ξ with respect to the probability measure P. At first, we give an elementary arithmetic property of modified ECF expansion in representation of real numbers, which was obtained by Fan et al. (2007). Proposition 2.1 (Fan et al., 2007, Proposition 2.3). Any real number x ∈ (0, 1] can be represented in the form of (1.1). Moreover, a number x ∈ (0, 1] has a finite modified ECF expansion (i.e. T n x = 0 for some n ≥ 1) if and only if x is rational. It is worth pointing out that not all positive real sequences can occur in the modified ECF expansion. Next we give the following definition, which describes the positive real sequences occurring in the modified ECF expansion. Definition 2.1. An n-block (d1 , d2 , . . . , dn ) is said to be admissible for the modified ECF expansion if there exists x ∈ (0, 1] such that dj (x) = dj for all 1 ≤ j ≤ n. An infinite sequence (d1 , d2 , . . . , dn , . . .) is called an admissible sequence if (d1 , d2 , . . . , dn ) is admissible for all n ≥ 1. The following proposition gives a characterization of all admissible sequences for the modified ECF expansion. Proposition 2.2. A sequence of positive integers (d1 , d2 , . . . , dn , . . .) is admissible for the modified ECF expansion if and only if for all n ≥ 1, dn+1 ≥ dn + 1. Proof. The necessity is obvious by the definition of dn and the modified ECF map T . To prove the sufficiency, for all n ≥ 1, we take 1 d1 + 1
x := ((d1 , d2 , . . . , dn )) = d1 +
.
. dn−1 + 1 d2 + . . + dn
Since d2 ≥ d1 + 1, we have 1 d1 + 1
1 d1
.
And hence the definition of dn , d1 (x) = d1 and Tx = ((d2 , d3 , . . . , dn )). Repeating the above procedure, we can get di (x) = di for all 1 ≤ i ≤ n. Thus, we get the desired result. Definition 2.2. Let (d1 , d2 , . . . , dn ) be an admissible sequence. We call B(d1 , d2 , . . . , dn ) = {x ∈ (0, 1) : d1 (x) = d1 , d2 (x) = d2 , . . . , dn (x) = dn }, the nth order cylinder. In other words, it is the set of points beginning with (d1 , . . . , dn ) in their modified ECF expansion. Suppose that {dn , n ≥ 1} is a sequence of positive integers. With the conventions p0 = 0, q0 = 1, p1 = 1 and q1 = d1 , let {pn , n ≥ 1} and {qn , n ≥ 1} be the sequence recursively defined by pn = dn pn−1 + (dn−1 + 1)pn−2
and
qn = dn qn−1 + (dn−1 + 1)qn−2 .
It is worth pointing out that qn = ((d1 , d2 , . . . , dn )) for all n ≥ 1. As for the usual continued fractions, we call n convergent of x ∈ (0, 1] in its modified ECF expansion (1.1). The following proposition is about the structure and the length of the cylinder. p
(2.5) pn (x) qn (x)
the nth
L. Fang / Statistics and Probability Letters 98 (2015) 98–106
101
Proposition 2.3 (Fan et al., 2007, Proposition 2.8). Let (d1 , d2 , . . . , dn ) be an admissible sequence, then B(d1 , d2 , . . . , dn ) is a p +p p half-open interval with two endpoints qn and qn +qn−1 . Hence that for all n ≥ 1, n
n−1
P(B(d1 , d2 , . . . , dn )) =
n
n−1
(di + 1)
i=1
qn (qn + qn−1 )
.
(2.6)
Although the digits sequence {dn , n ≥ 1} does not form a Markov chain, we have the following proposition. Proposition 2.4. Let {dn , n ≥ 1} be the digits sequence occurring in the modified ECF expansion. Then for all k ≥ j + 1 and j ≥ 1, we have P(d1 = j) =
1
(2.7)
j(j + 1)
and j+1 j+3 ≤ P(dn+1 = k|dn = j) ≤ . (k + 1)(k + 2) k(k + 1)
(2.8)
Proof. (2.7) is obvious by taking n = 1 in (2.6). For two integers 1 ≤ a ≤ b and real number 0 ≤ y < 1, we define
Φ (a, b, y) =
a(1 + y) . (b + ay)(b + 1 + ay)
By (2.6), we deduce that for all n ≥ 1, P(B(d1 , d2 , . . . , dn , dn+1 )) P(B(d1 , d2 , . . . , dn ))
= Φ (dn + 1, dn+1 , yn ) with yn =
qn−1 qn
.
(2.9)
In view of (2.5), we know that dn ≤ dn+1 and dn qn−1 ≤ qn for all n ≥ 1. And hence that by (2.9), dn + 1 dn + 3 ≤ Φ (dn + 1, dn+1 , yn ) ≤ . (dn+1 + 1)(dn+1 + 2) dn+1 (dn+1 + 1) Combining (2.9) and (2.10), then the inequalities (2.8) are established.
(2.10)
Let {Xn , n ≥ 1} be a sequence of the real-valued random variables defined on probability space (Ω , F , P). For any λ ∈ R and all n ≥ 1, define the logarithmic moment generating function of Xn ,
Λn (λ) := log E(eλXn ). In the proofs of Theorems 1.1 and 1.2, we will make use of the following Gärtner–Ellis Theorem, see e.g. Dembo and Zeitouni (1998). Proposition 2.5 (Gärtner–Ellis Theorem). Assume that Λ(λ) := limn→∞ (1/n)Λn (nλ) exists over the domain DΛ := {λ ∈ R :
Λ(λ) < ∞} and also satisfies the following conditions: 1. 2. 3. 4.
The interior of DΛ contains the origin. Λ(·) is a lower semicontinuous function. Λ(·) is differentiable throughout the interior of DΛ . limn→∞ |Λ′ (λn )| = ∞ whenever {λn , n ≥ 1} is a sequence in the interior of DΛ converging to a boundary point of interior of DΛ .
Then the sequence {Xn , n ≥ 1} satisfies a LDP with speed n and good rate function I (x) = sup{λx − Λ(λ)}, λ∈R
∀x ∈ R.
Remark 2. When 1/n is replaced by a positive real number sequence an with an → 0 as n → ∞ and the function Λ(·) is properly modified, the results of Gärtner–Ellis Theorem are also valid. 3. Proof of theorems In this section, we will give the proofs of Theorems 1.1 and 1.2. The following Lemma 3.1 is the key lemma in the proof of Theorems 1.1 and 1.2.
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Lemma 3.1. Let θ < 1. Then for all j ≥ 1, we have ∞
θ
j+3
k
k(k + 1) k=j+1
3
≤ 1+
j
1
·
j
1−θ
and ∞
θ
j+1
k
(k + 1)(k + 2) k=j+1
≥
j
j
1+
j+2
θ−1
1
·
j
1 1−θ
.
Proof. Let θ < 1. Notice that for any j ≥ 1, ∞
j
1 x2−θ
1
dx =
· jθ −1
1−θ
(3.11)
and ∞
kθ
k=j+1
k(k + 1)
∞
≤
1
≤
k2−θ
k=j+1
∞
1
x2−θ
j
dx.
(3.12)
Therefore, by (3.11) and (3.12), we have that for all j ≥ 1 ∞
j+3
θ k
k(k + 1) k=j+1
j
kθ
∞ j+3
=
jθ
3 ≤ 1+ ·
k(k + 1) k=j+1
1 1−θ
j
.
On the other hand, note that for any j ≥ 1, ∞ k=j+1
1 k2−θ
∞
≥
j+1
1
dx =
x2−θ
1 1−θ
· (j + 1)θ−1
(3.13)
and ∞
j+1
θ
k=j+1
(k + 1)(k + 2)
j
k
=
=
kθ
∞ j+1
jθ
k=j+1
(k + 1)(k + 2)
∞ j+1
jθ
1
k2−θ k=j+1
·
k k+1
·
k k+2
.
Since k/(k + 1) ≥ j/(j + 1) for all k ≥ j, in view of (3.13) and (3.14), we deduce that for all j ≥ 1, ∞ k=j+1
θ
j+1
k
(k + 1)(k + 2)
≥
j
≥
∞ j+1
jθ
1
k=j+1
1 1−θ
·
·
k2−θ
j
j j+1
j+2
1+
1 j
·
j j+2
θ−1
.
3.1. Proof of Theorem 1.1 To prove Theorem 1.1, we also need the following Lemma 3.2. Lemma 3.2. Let {dn , n ≥ 1} be the digits sequence of the modified ECF expansion. Then lim
n→∞
1 n
log E(dθn ) =
log
1
1−θ
if θ < 1, if θ ≥ 1.
+∞
Proof. For any θ ≥ 1, note that for all n ≥ 1, dn+1 > dn with d1 ≥ 1 and (2.7), so for any n ≥ 1 E(dθn ) ≥ E(dθ1 ) =
∞ k=1
kθ ·
1 k(k + 1)
Therefore, for any θ ≥ 1, limn→∞
1 n
= +∞.
log E (dθn ) = +∞.
(3.14)
L. Fang / Statistics and Probability Letters 98 (2015) 98–106
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Let θ < 1. Since for all n ≥ 1, dn+1 ≥ dn + 1 with d1 ≥ 1, we have that dn ≥ n for all n ≥ 1. By the definition of expectation, we deduce that for all n ≥ 1, E(dθn ) =
∞
P(dn = k) · kθ
k=n
=
∞ k −1
P(dn = k|dn−1 = j)P(dn−1 = j) · kθ
k=n j=n−1
=
∞
∞
P(dn−1 = j) · jθ
j=n−1
P(dn = k|dn−1 = j)
θ k j
k=j+1
,
(3.15)
where the second equality follows from the conditional probability. By Proposition 2.4 and Lemma 3.1, for all j ≥ 1, we have 1
·
1−θ
j
1+
j+2
1
θ−1
∞
≤
j
P(dn = k|dn−1 = j)
θ k j
k=j+1
3 ≤ 1+ · j
1 1−θ
.
(3.16)
Therefore, in view of (3.15) and the first inequality of (3.16), for all n ≥ 1, we have θ
E(dn ) ≥
∞
θ
P(dn−1 = j) · j ·
j=n−1
≥
∞
P(dn−1 = j) · jθ ·
j=n−1
≥ ······ ∞ ≥ P(d1 = j) · jθ · j =1
1 1−θ 1 1−θ
·
·
j j+2
j+1
n−1
n−1
n+1
n−1
1
·
1−θ
1−θ
j
1−θ
n
1−θ
2
1
n(n + 1)
(3.17)
n
j
where the second inequality follows from the fact j+1 ≥ i+i 1 for all j ≥ i. In view of (3.17), we deduce that for all n ≥ 1, θ
θ −3
E(dn ) ≥ Mn where M =
1−θ
jθ j=1 j(j+1)
∞
n −1
1
,
(3.18)
is a positive constant. Thus, we complete the lower bounded estimate of E(dθn ).
Next, we give the upper bounded estimate of E(dθn ). By (3.15) and the second inequality of (3.16), for all n ≥ 1, we obtain θ
E(dn ) ≤
∞
θ
P(dn−1 = j) · j ·
j=n−1
≤
∞
θ
P(dn−1 = j) · j ·
j=n−1
≤ ······ ∞ ≤ P(d1 = j) · jθ · j =1
1 1−θ
1 1−θ
1
· 1+ ·
j
n+2
n−1
n−1 ·
1−θ
3
n(n + 1)(n + 2)
where the second inequality follows from the fact 1 + E(dθn ) ≤ Mn3
∞
n −1
1 1−θ
(3.19)
6 1 j
≤1+
1 i
for all j ≥ i. Thus, for all n ≥ 1, we have that by (3.19)
,
(3.20)
jθ
where M = j=1 j(j+1) is a positive constant. By (3.18) and (3.20), we deduce that lim inf n→∞
1 n
log E(dθn ) ≥ lim inf
log M n
n→∞
= log
1 1−θ
+ (θ − 3)
log n n
+
n−1 n
log
1 1−θ
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L. Fang / Statistics and Probability Letters 98 (2015) 98–106
and 1
lim sup
n
n→∞
log E(dθn ) ≤ lim sup
log M n
n→∞
= log
1 1−θ
+
3 log n n
+
n−1 n
log
1 1−θ
.
That is, for θ < 1, lim
n→∞
1 n
θ
log E(dn ) = log
1 1−θ
.
Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1. Note that
Λ(θ ) = −θ + lim
1
n→∞
log E(dθn ) =
n
−θ − log(1 − θ ) +∞
if θ < 1, if θ ≥ 1.
It is not difficult check thatΛ(·) satisfies all the conditions of Proposition 2.5. By Gärtner–Ellis theorem, we know that logto dn −n the sequence , n ≥ 1 satisfies a large deviation principle with speed n and good rate function n I (x) = sup {θ x − Λ(θ )} = sup {θ x + θ + log(1 − θ )} = θ∈R
θ<1
x − log(1 + x) +∞
if x > −1, if x ≤ −1.
3.2. Proof of Theorem 1.2 As an application of Proposition 2.4 and Lemma 3.1, we obtain the following lemma. Lemma 3.3. Let θ ∈ (−1, 1). Then for any j ≥ 1, we have 1 1−θ
·
3
j
≤
j+2
∞
P(dn = k|dn−1 = j)
θ k j
k=j+1
≤
j+3 j
·
1 1−θ
.
(3.21)
Now we are going to give the proof of Theorem 1.2. Proof of Theorem 1.2. For any λ ∈ R, let us consider the following logarithmic moment generating function of
log dn −n , an
log dn − n . Λn (λ) = log E exp λ · an
From the Gärtner–Ellis theorem, in order to obtain the desired result, it suffices to show that for any λ ∈ R,
Λ(λ) = lim
n→∞
n
Λn 2
a2n
an
n
λ =
λ2 2
.
That is, lim
n
n→∞ a2 n
log E exp
a
n
n
(log dn − n) λ
=
λ2 2
.
(3.22)
For any fixed real λ and for all n ≥ 1, let
θn := θn (λ) =
an n
λ and Υn (λ) = E(exp{θn (log dn − n)}).
Then in view of (1.4), it is clear that θn → 0 as n → ∞ and Υn (λ) can be rewritten as
Υn (λ) = e−nθn E(dθnn ).
(3.23)
To get (3.22), we only need to estimate the expectation E(dθnn ). Since θn → 0 as n → ∞, there exists N > 0 such that n ≥ N, we have θn ∈ (− 12 , 12 ). Now we will give the lower and upper bounded estimate of E(dθnn ). By (3.15) and the first
L. Fang / Statistics and Probability Letters 98 (2015) 98–106
105
inequality of (3.21), note that for all n ≥ N, ∞
E(dθnn ) =
∞
P(dn−1 = j) · jθn
j=n−1
k=j+1
∞
≥
P(dn−1 = j) · jθn ·
P(dn = k|dn−1 = j)
n−1
3 ·
n+1
j=n−1
θn k j
1 1 − θn
≥ ······ n −1 3 ∞ 1 2 n−2 n−1 1 . ≥ P(d1 = j) · jθn · · · ··· · · 3 4 n n+1 1 − θn j =1 Let M =
∞
4 j=1 j3/2 (j+1) ,
then M is a positive constant and we deduce that for all n ≥ N,
M
E(dθnn ) ≥
n
1
(n + 1)6
.
1 − θn
(3.24)
On the other hand, in view of (3.15) and the second inequality of (3.21), we have for all n ≥ N, ∞
θn
E(dn ) =
∞
θn
P(dn−1 = j) · j
j=n−1
k=j+1
∞
≤
P(dn−1 = j) · jθn ·
P(dn = k|dn−1 = j)
n+2 n−1
j=n−1
·
θn k j
1 1 − θn
≤ ······ n−1 ∞ 4 5 n+1 n+2 1 P(d1 = j) · jθn · ≤ · · ··· · · . 1 2 n−2 n−1 1 − θn j =1 Let M ′ =
3 2
∞
1 j=1 j1/2 (j+1) ,
E(dθnn ) ≤ M ′ n3
then M ′ is a positive constant and we obtain that for all n ≥ N,
n
1 1 − θn
.
(3.25)
Combining (1.4), (3.23), (3.24) and (3.25), we deduce that by Taylor expansion lim inf n→∞
n a2n
log Υn (λ) = lim inf
n2 a2n
n→∞
≥ lim inf n→∞
n a2n
(−θn ) + (−θn ) +
n
log E(dn ) θn
a2n n a2n
log M −
6n log(n + 1) a2n
−
n2 a2n
log(1 − θn )
λ2
=
(3.26)
2
and lim sup n→∞
n a2n
log Υn (λ) = lim sup
n→∞
≤ lim sup n→∞
=
λ2 2
n2
n
an
a2n
n2
n
an
a2n
(−θn ) + 2 (−θn ) + 2
log E(dθnn ) log M ′ +
3n log n a2n
−
n a2n
log(1 − θn )
.
(3.27)
By (3.26) and (3.27), Eq. (3.22) is established. By Gärtner–Ellis theorem, we know that the sequence { satisfies an MDP with speed n
−1 2
an and good rate function
I (x) = sup {λx − Λ(λ)} = λ∈R
x2 2
,
∀ x ∈ R.
log dn −n an
, n ≥ 1}
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Acknowledgments We thank Dr. Rui Kuang and Dr. Yuanyuan Ren for their useful discussions. We would also like to thank the anonymous referee for the detailed and helpful comments and suggestions. References Dembo, A., Zeitouni, O., 1998. Large Deviations Techniques and Applications, second ed. Springer, New York. Erdős, P., Rényi, A., Szüsz, P., 1958. On Engel’s and Sylvester’s series. Ann. Univ. Sci. Budapest. Eötvös. Sect. Math. 1, 7–32. Fan, A., Wang, B., Wu, J., 2007. Arithmetic and metric properties of Oppenheim continued fraction expansions. J. Number Theory 127, 64–82. Hartono, Y., Kraaikamp, C., Schweiger, F., 2002. Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients. J. Théor. Nombres Bordeaux 14, 497–516. Hu, W., 2015. Moderate deviation principles for Engel’s, Sylvester’s series and Cantor’s products. Statist. Probab. Lett. 96, 247–254. Mehrdad, B., Zhu, L., 2013. Moderate and large deviations for Erdős–Kac theorem. Arxiv Preprint. arXiv:1311.6180. Rényi, A., 1962. A new approach to the theory of Engel’s series. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 5, 25–32. Touchette, H., 2009. The large deviation approach to statistical mechanics. Phys. Rep. 478, 1–69. Varadhan, S.R.S., 1984. Large Deviations and Applications. SIAM, Philadelphia. Zhu, L., 2014. On the large deviations for Engel’s, Sylvester’s series and Cantor’s products. Electron. Comm. Probab. 19, 1–9.
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Large and moderate deviations for modified Engel continued fractions Lulu Fang Department of Mathematics, South China University of Technology, Guangzhou 510640, PR China
article
abstract
info
Article history: Received 28 November 2014 Received in revised form 9 December 2014 Accepted 11 December 2014 Available online 24 December 2014
In this paper, we consider the large and moderate deviation principles for modified Engel continued fractions which is a representation of real numbers in number theory. © 2014 Elsevier B.V. All rights reserved.
MSC: 60F10 11A67 11K55 Keywords: Large deviations Moderate deviations Modified Engel continued fractions Good rate function
1. Introduction Hartono et al. (2002) introduced a new continued fraction algorithm with non-decreasing partial quotients, called the Engel continued fraction (ECF for short) expansion. The name of this new continued fraction expansion is originated from the Engel series expansion considered by Erdős et al. (1958), which is a classical representation of real numbers in number theory. Similarly, we can obtain another new continued fractions named the modified ECF expansion by borrowing from the modified Engel series expansion which was studied by Rényi (1962). Here we give the algorithm of the generation of modified ECF expansion. Let T : (0, 1] −→ (0, 1] be the modified ECF map, which is given by 1 Tx = 1 x
+1
·
1 x
1
−
x
,
where [x] denotes the greatest integer not exceeding x. For any real number x ∈ (0, 1] and all n ≥ 1, let d1 (x) = [1/x] and dn+1 (x) = d1 (T n x), where T n denotes the nth iterate of T . Then for every x ∈ (0, 1], the modified ECF map T can generate a continued fraction expansion in the form of x= d1 +
1 d1 + 1
,
. dn−1 + 1 d2 + . . + . dn + . .
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2014.12.015 0167-7152/© 2014 Elsevier B.V. All rights reserved.
(1.1)
L. Fang / Statistics and Probability Letters 98 (2015) 98–106
99
where dn ≥ 1 are positive integers and dn+1 ≥ dn + 1 for all n ≥ 1. The representation (1.1) is said to be the modified ECF expansion of x and dn (x), n ≥ 1 are called the digits of the modified ECF expansion of x. Sometimes we write the form (1.1) as x = ((d1 , d2 , . . . , dn , . . .)). In fact, this modified ECF expansion is a special case of the Oppenheim continued fractions which was studied by Fan et al. (2007). Moreover, some arithmetic and statistical properties of the modified ECF expansion, such as the representation of rational numbers, law of large numbers and central limit theorem were also discussed by Fan et al. (2007). Now we turn to introducing the large and moderate deviation principles. Let {Xn , n ≥ 1} be a sequence of the realvalued random variables defined on the probability space (Ω , F , P). A function I : R → [0, ∞] is called a good rate function if it is lower semicontinuous and has compact level sets. Let {λn , n ≥ 1} be a positive real number sequence with limn→∞ λn = ∞. We say that the sequence {Xn , n ≥ 1} satisfies a large deviation principle (LDP for short) with speed λn and good rate function I under P, if for any Borel set Γ , we have
− inf◦ I (x) ≤ lim inf x∈Γ
n→∞
1
λn
log P(Xn ∈ Γ ) ≤ lim sup n→∞
1
λn
log P(Xn ∈ Γ ) ≤ − sup I (x),
(1.2)
x∈Γ
where Γ ◦ and Γ denotes the interior and the closure of Γ respectively. Formally there is no distinction between the large deviation principle and the moderate deviation principle (MDP for short). Usually LDP characterizes the convergence speed of the law of large numbers, while MDP describes the speed of convergence speed between the law of large numbers and the central limit theorem. We refer the reader to Dembo and Zeitouni (1998), Touchette (2009) and Varadhan (1984) for general backgrounds of large and moderate deviations. In this paper, we denote by (Ω , F , P) a probability space, where Ω = (0, 1], F is the Borel σ -algebra on Ω and P denotes the Lebesgue measure on (Ω , F ). Fan et al. (2007) showed a strong law of large numbers for the digits sequence {dn , n ≥ 1} occurring in the modified ECF expansion, i.e., for almost all x ∈ (0, 1], we have lim
n→∞
1
log dn (x) = 1.
n
And also proved that the central limit theorem holds for the digits sequence {dn , n ≥ 1}. That is, for every y ∈ R, we have
lim P
log dn − n
√
n→∞
1 ≤y = √
2π
n
y
2 e−t /2 dt .
−∞
A natural question is to consider the probability of the event that leads to the study of large deviations for the modified ECF expansion.
log dn n
deviates away from its ergodic mean of 1? This
Theorem 1.1. Let {dn , n ≥ 1} be the digits sequence of the modified ECF expansion. Then speed n and good rate function I (x) =
x − log(x + 1)
+∞
log dn −n n
, n ≥ 1 satisfies a LDP with
if x > −1 otherwise,
(1.3)
under P. As a complement of Theorem 1.1, we give the following MDP result for the modified ECF expansion. Theorem 1.2. Let {dn , n ≥ 1} be the digits sequence of the modified ECF expansion and {an , n ≥ 1} be a positive sequence satisfying an → ∞, Then
log dn −n an
√
an
n log n
→ ∞ and
an n
→ 0.
(1.4)
, n ≥ 1 satisfies an MDP with speed n−1 a2n and good rate function I (x) = x2 /2 under P. √
Remark 1. We may obtain the result if the second condition of an in (1.4) is replaced by an / n → ∞ as n → ∞. However, n we need the condition √nalog → ∞ to avoid the technical difficulties. The main reason is that we cannot find the finer n estimates when we make use of the Gärtner–Ellis Theorem (see Dembo and Zeitouni, 1998, p. 44, Theorem 2.3.6). As an application of Theorem 1.2, we obtain the following corollary. Corollary 1. Let {dn , n ≥ 1} be the digits sequence of the modified ECF expansion and an = np with p ∈ ( 12 , 1). Then
log dn −n an
, n ≥ 1 satisfies an MDP with speed n−1 a2n and good rate function I (x) = x2 /2 under P.
Recently, some authors began to consider the large and moderate deviations related to number theory. For example, Mehrdad and Zhu (2013) established the large and moderate deviations for Erdős–Kac theorem, which is a celebrated result
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L. Fang / Statistics and Probability Letters 98 (2015) 98–106
about the number of distinct prime factors of a uniformly chosen random integer in number theory. Zhu (2014) and Hu (2015) studied respectively the large deviations and moderate deviations for Engel, Sylvester and Cantor product series, which are the classical representations of real numbers in number theory. Note that the digits sequence occurring in the Engel, Sylvester and Cantor product series forms a Markov chain and the proofs of Zhu (2014) and Hu (2015) rely on this property, while the digits sequence of modified ECF expansion is not a Markov chain. We overcome this difficulty by using a weak property (i.e. Proposition 2.4) in Section 2. 2. Preliminary In this section, we recall some definitions and some arithmetic and metric properties of the modified ECF expansion. We denote (Ω , F , P) as a probability space and use the notation E(ξ ) to denote the expectation of the random variable ξ with respect to the probability measure P. At first, we give an elementary arithmetic property of modified ECF expansion in representation of real numbers, which was obtained by Fan et al. (2007). Proposition 2.1 (Fan et al., 2007, Proposition 2.3). Any real number x ∈ (0, 1] can be represented in the form of (1.1). Moreover, a number x ∈ (0, 1] has a finite modified ECF expansion (i.e. T n x = 0 for some n ≥ 1) if and only if x is rational. It is worth pointing out that not all positive real sequences can occur in the modified ECF expansion. Next we give the following definition, which describes the positive real sequences occurring in the modified ECF expansion. Definition 2.1. An n-block (d1 , d2 , . . . , dn ) is said to be admissible for the modified ECF expansion if there exists x ∈ (0, 1] such that dj (x) = dj for all 1 ≤ j ≤ n. An infinite sequence (d1 , d2 , . . . , dn , . . .) is called an admissible sequence if (d1 , d2 , . . . , dn ) is admissible for all n ≥ 1. The following proposition gives a characterization of all admissible sequences for the modified ECF expansion. Proposition 2.2. A sequence of positive integers (d1 , d2 , . . . , dn , . . .) is admissible for the modified ECF expansion if and only if for all n ≥ 1, dn+1 ≥ dn + 1. Proof. The necessity is obvious by the definition of dn and the modified ECF map T . To prove the sufficiency, for all n ≥ 1, we take 1 d1 + 1
x := ((d1 , d2 , . . . , dn )) = d1 +
.
. dn−1 + 1 d2 + . . + dn
Since d2 ≥ d1 + 1, we have 1 d1 + 1
1 d1
.
And hence the definition of dn , d1 (x) = d1 and Tx = ((d2 , d3 , . . . , dn )). Repeating the above procedure, we can get di (x) = di for all 1 ≤ i ≤ n. Thus, we get the desired result. Definition 2.2. Let (d1 , d2 , . . . , dn ) be an admissible sequence. We call B(d1 , d2 , . . . , dn ) = {x ∈ (0, 1) : d1 (x) = d1 , d2 (x) = d2 , . . . , dn (x) = dn }, the nth order cylinder. In other words, it is the set of points beginning with (d1 , . . . , dn ) in their modified ECF expansion. Suppose that {dn , n ≥ 1} is a sequence of positive integers. With the conventions p0 = 0, q0 = 1, p1 = 1 and q1 = d1 , let {pn , n ≥ 1} and {qn , n ≥ 1} be the sequence recursively defined by pn = dn pn−1 + (dn−1 + 1)pn−2
and
qn = dn qn−1 + (dn−1 + 1)qn−2 .
It is worth pointing out that qn = ((d1 , d2 , . . . , dn )) for all n ≥ 1. As for the usual continued fractions, we call n convergent of x ∈ (0, 1] in its modified ECF expansion (1.1). The following proposition is about the structure and the length of the cylinder. p
(2.5) pn (x) qn (x)
the nth
L. Fang / Statistics and Probability Letters 98 (2015) 98–106
101
Proposition 2.3 (Fan et al., 2007, Proposition 2.8). Let (d1 , d2 , . . . , dn ) be an admissible sequence, then B(d1 , d2 , . . . , dn ) is a p +p p half-open interval with two endpoints qn and qn +qn−1 . Hence that for all n ≥ 1, n
n−1
P(B(d1 , d2 , . . . , dn )) =
n
n−1
(di + 1)
i=1
qn (qn + qn−1 )
.
(2.6)
Although the digits sequence {dn , n ≥ 1} does not form a Markov chain, we have the following proposition. Proposition 2.4. Let {dn , n ≥ 1} be the digits sequence occurring in the modified ECF expansion. Then for all k ≥ j + 1 and j ≥ 1, we have P(d1 = j) =
1
(2.7)
j(j + 1)
and j+1 j+3 ≤ P(dn+1 = k|dn = j) ≤ . (k + 1)(k + 2) k(k + 1)
(2.8)
Proof. (2.7) is obvious by taking n = 1 in (2.6). For two integers 1 ≤ a ≤ b and real number 0 ≤ y < 1, we define
Φ (a, b, y) =
a(1 + y) . (b + ay)(b + 1 + ay)
By (2.6), we deduce that for all n ≥ 1, P(B(d1 , d2 , . . . , dn , dn+1 )) P(B(d1 , d2 , . . . , dn ))
= Φ (dn + 1, dn+1 , yn ) with yn =
qn−1 qn
.
(2.9)
In view of (2.5), we know that dn ≤ dn+1 and dn qn−1 ≤ qn for all n ≥ 1. And hence that by (2.9), dn + 1 dn + 3 ≤ Φ (dn + 1, dn+1 , yn ) ≤ . (dn+1 + 1)(dn+1 + 2) dn+1 (dn+1 + 1) Combining (2.9) and (2.10), then the inequalities (2.8) are established.
(2.10)
Let {Xn , n ≥ 1} be a sequence of the real-valued random variables defined on probability space (Ω , F , P). For any λ ∈ R and all n ≥ 1, define the logarithmic moment generating function of Xn ,
Λn (λ) := log E(eλXn ). In the proofs of Theorems 1.1 and 1.2, we will make use of the following Gärtner–Ellis Theorem, see e.g. Dembo and Zeitouni (1998). Proposition 2.5 (Gärtner–Ellis Theorem). Assume that Λ(λ) := limn→∞ (1/n)Λn (nλ) exists over the domain DΛ := {λ ∈ R :
Λ(λ) < ∞} and also satisfies the following conditions: 1. 2. 3. 4.
The interior of DΛ contains the origin. Λ(·) is a lower semicontinuous function. Λ(·) is differentiable throughout the interior of DΛ . limn→∞ |Λ′ (λn )| = ∞ whenever {λn , n ≥ 1} is a sequence in the interior of DΛ converging to a boundary point of interior of DΛ .
Then the sequence {Xn , n ≥ 1} satisfies a LDP with speed n and good rate function I (x) = sup{λx − Λ(λ)}, λ∈R
∀x ∈ R.
Remark 2. When 1/n is replaced by a positive real number sequence an with an → 0 as n → ∞ and the function Λ(·) is properly modified, the results of Gärtner–Ellis Theorem are also valid. 3. Proof of theorems In this section, we will give the proofs of Theorems 1.1 and 1.2. The following Lemma 3.1 is the key lemma in the proof of Theorems 1.1 and 1.2.
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L. Fang / Statistics and Probability Letters 98 (2015) 98–106
Lemma 3.1. Let θ < 1. Then for all j ≥ 1, we have ∞
θ
j+3
k
k(k + 1) k=j+1
3
≤ 1+
j
1
·
j
1−θ
and ∞
θ
j+1
k
(k + 1)(k + 2) k=j+1
≥
j
j
1+
j+2
θ−1
1
·
j
1 1−θ
.
Proof. Let θ < 1. Notice that for any j ≥ 1, ∞
j
1 x2−θ
1
dx =
· jθ −1
1−θ
(3.11)
and ∞
kθ
k=j+1
k(k + 1)
∞
≤
1
≤
k2−θ
k=j+1
∞
1
x2−θ
j
dx.
(3.12)
Therefore, by (3.11) and (3.12), we have that for all j ≥ 1 ∞
j+3
θ k
k(k + 1) k=j+1
j
kθ
∞ j+3
=
jθ
3 ≤ 1+ ·
k(k + 1) k=j+1
1 1−θ
j
.
On the other hand, note that for any j ≥ 1, ∞ k=j+1
1 k2−θ
∞
≥
j+1
1
dx =
x2−θ
1 1−θ
· (j + 1)θ−1
(3.13)
and ∞
j+1
θ
k=j+1
(k + 1)(k + 2)
j
k
=
=
kθ
∞ j+1
jθ
k=j+1
(k + 1)(k + 2)
∞ j+1
jθ
1
k2−θ k=j+1
·
k k+1
·
k k+2
.
Since k/(k + 1) ≥ j/(j + 1) for all k ≥ j, in view of (3.13) and (3.14), we deduce that for all j ≥ 1, ∞ k=j+1
θ
j+1
k
(k + 1)(k + 2)
≥
j
≥
∞ j+1
jθ
1
k=j+1
1 1−θ
·
·
k2−θ
j
j j+1
j+2
1+
1 j
·
j j+2
θ−1
.
3.1. Proof of Theorem 1.1 To prove Theorem 1.1, we also need the following Lemma 3.2. Lemma 3.2. Let {dn , n ≥ 1} be the digits sequence of the modified ECF expansion. Then lim
n→∞
1 n
log E(dθn ) =
log
1
1−θ
if θ < 1, if θ ≥ 1.
+∞
Proof. For any θ ≥ 1, note that for all n ≥ 1, dn+1 > dn with d1 ≥ 1 and (2.7), so for any n ≥ 1 E(dθn ) ≥ E(dθ1 ) =
∞ k=1
kθ ·
1 k(k + 1)
Therefore, for any θ ≥ 1, limn→∞
1 n
= +∞.
log E (dθn ) = +∞.
(3.14)
L. Fang / Statistics and Probability Letters 98 (2015) 98–106
103
Let θ < 1. Since for all n ≥ 1, dn+1 ≥ dn + 1 with d1 ≥ 1, we have that dn ≥ n for all n ≥ 1. By the definition of expectation, we deduce that for all n ≥ 1, E(dθn ) =
∞
P(dn = k) · kθ
k=n
=
∞ k −1
P(dn = k|dn−1 = j)P(dn−1 = j) · kθ
k=n j=n−1
=
∞
∞
P(dn−1 = j) · jθ
j=n−1
P(dn = k|dn−1 = j)
θ k j
k=j+1
,
(3.15)
where the second equality follows from the conditional probability. By Proposition 2.4 and Lemma 3.1, for all j ≥ 1, we have 1
·
1−θ
j
1+
j+2
1
θ−1
∞
≤
j
P(dn = k|dn−1 = j)
θ k j
k=j+1
3 ≤ 1+ · j
1 1−θ
.
(3.16)
Therefore, in view of (3.15) and the first inequality of (3.16), for all n ≥ 1, we have θ
E(dn ) ≥
∞
θ
P(dn−1 = j) · j ·
j=n−1
≥
∞
P(dn−1 = j) · jθ ·
j=n−1
≥ ······ ∞ ≥ P(d1 = j) · jθ · j =1
1 1−θ 1 1−θ
·
·
j j+2
j+1
n−1
n−1
n+1
n−1
1
·
1−θ
1−θ
j
1−θ
n
1−θ
2
1
n(n + 1)
(3.17)
n
j
where the second inequality follows from the fact j+1 ≥ i+i 1 for all j ≥ i. In view of (3.17), we deduce that for all n ≥ 1, θ
θ −3
E(dn ) ≥ Mn where M =
1−θ
jθ j=1 j(j+1)
∞
n −1
1
,
(3.18)
is a positive constant. Thus, we complete the lower bounded estimate of E(dθn ).
Next, we give the upper bounded estimate of E(dθn ). By (3.15) and the second inequality of (3.16), for all n ≥ 1, we obtain θ
E(dn ) ≤
∞
θ
P(dn−1 = j) · j ·
j=n−1
≤
∞
θ
P(dn−1 = j) · j ·
j=n−1
≤ ······ ∞ ≤ P(d1 = j) · jθ · j =1
1 1−θ
1 1−θ
1
· 1+ ·
j
n+2
n−1
n−1 ·
1−θ
3
n(n + 1)(n + 2)
where the second inequality follows from the fact 1 + E(dθn ) ≤ Mn3
∞
n −1
1 1−θ
(3.19)
6 1 j
≤1+
1 i
for all j ≥ i. Thus, for all n ≥ 1, we have that by (3.19)
,
(3.20)
jθ
where M = j=1 j(j+1) is a positive constant. By (3.18) and (3.20), we deduce that lim inf n→∞
1 n
log E(dθn ) ≥ lim inf
log M n
n→∞
= log
1 1−θ
+ (θ − 3)
log n n
+
n−1 n
log
1 1−θ
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L. Fang / Statistics and Probability Letters 98 (2015) 98–106
and 1
lim sup
n
n→∞
log E(dθn ) ≤ lim sup
log M n
n→∞
= log
1 1−θ
+
3 log n n
+
n−1 n
log
1 1−θ
.
That is, for θ < 1, lim
n→∞
1 n
θ
log E(dn ) = log
1 1−θ
.
Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1. Note that
Λ(θ ) = −θ + lim
1
n→∞
log E(dθn ) =
n
−θ − log(1 − θ ) +∞
if θ < 1, if θ ≥ 1.
It is not difficult check thatΛ(·) satisfies all the conditions of Proposition 2.5. By Gärtner–Ellis theorem, we know that logto dn −n the sequence , n ≥ 1 satisfies a large deviation principle with speed n and good rate function n I (x) = sup {θ x − Λ(θ )} = sup {θ x + θ + log(1 − θ )} = θ∈R
θ<1
x − log(1 + x) +∞
if x > −1, if x ≤ −1.
3.2. Proof of Theorem 1.2 As an application of Proposition 2.4 and Lemma 3.1, we obtain the following lemma. Lemma 3.3. Let θ ∈ (−1, 1). Then for any j ≥ 1, we have 1 1−θ
·
3
j
≤
j+2
∞
P(dn = k|dn−1 = j)
θ k j
k=j+1
≤
j+3 j
·
1 1−θ
.
(3.21)
Now we are going to give the proof of Theorem 1.2. Proof of Theorem 1.2. For any λ ∈ R, let us consider the following logarithmic moment generating function of
log dn −n , an
log dn − n . Λn (λ) = log E exp λ · an
From the Gärtner–Ellis theorem, in order to obtain the desired result, it suffices to show that for any λ ∈ R,
Λ(λ) = lim
n→∞
n
Λn 2
a2n
an
n
λ =
λ2 2
.
That is, lim
n
n→∞ a2 n
log E exp
a
n
n
(log dn − n) λ
=
λ2 2
.
(3.22)
For any fixed real λ and for all n ≥ 1, let
θn := θn (λ) =
an n
λ and Υn (λ) = E(exp{θn (log dn − n)}).
Then in view of (1.4), it is clear that θn → 0 as n → ∞ and Υn (λ) can be rewritten as
Υn (λ) = e−nθn E(dθnn ).
(3.23)
To get (3.22), we only need to estimate the expectation E(dθnn ). Since θn → 0 as n → ∞, there exists N > 0 such that n ≥ N, we have θn ∈ (− 12 , 12 ). Now we will give the lower and upper bounded estimate of E(dθnn ). By (3.15) and the first
L. Fang / Statistics and Probability Letters 98 (2015) 98–106
105
inequality of (3.21), note that for all n ≥ N, ∞
E(dθnn ) =
∞
P(dn−1 = j) · jθn
j=n−1
k=j+1
∞
≥
P(dn−1 = j) · jθn ·
P(dn = k|dn−1 = j)
n−1
3 ·
n+1
j=n−1
θn k j
1 1 − θn
≥ ······ n −1 3 ∞ 1 2 n−2 n−1 1 . ≥ P(d1 = j) · jθn · · · ··· · · 3 4 n n+1 1 − θn j =1 Let M =
∞
4 j=1 j3/2 (j+1) ,
then M is a positive constant and we deduce that for all n ≥ N,
M
E(dθnn ) ≥
n
1
(n + 1)6
.
1 − θn
(3.24)
On the other hand, in view of (3.15) and the second inequality of (3.21), we have for all n ≥ N, ∞
θn
E(dn ) =
∞
θn
P(dn−1 = j) · j
j=n−1
k=j+1
∞
≤
P(dn−1 = j) · jθn ·
P(dn = k|dn−1 = j)
n+2 n−1
j=n−1
·
θn k j
1 1 − θn
≤ ······ n−1 ∞ 4 5 n+1 n+2 1 P(d1 = j) · jθn · ≤ · · ··· · · . 1 2 n−2 n−1 1 − θn j =1 Let M ′ =
3 2
∞
1 j=1 j1/2 (j+1) ,
E(dθnn ) ≤ M ′ n3
then M ′ is a positive constant and we obtain that for all n ≥ N,
n
1 1 − θn
.
(3.25)
Combining (1.4), (3.23), (3.24) and (3.25), we deduce that by Taylor expansion lim inf n→∞
n a2n
log Υn (λ) = lim inf
n2 a2n
n→∞
≥ lim inf n→∞
n a2n
(−θn ) + (−θn ) +
n
log E(dn ) θn
a2n n a2n
log M −
6n log(n + 1) a2n
−
n2 a2n
log(1 − θn )
λ2
=
(3.26)
2
and lim sup n→∞
n a2n
log Υn (λ) = lim sup
n→∞
≤ lim sup n→∞
=
λ2 2
n2
n
an
a2n
n2
n
an
a2n
(−θn ) + 2 (−θn ) + 2
log E(dθnn ) log M ′ +
3n log n a2n
−
n a2n
log(1 − θn )
.
(3.27)
By (3.26) and (3.27), Eq. (3.22) is established. By Gärtner–Ellis theorem, we know that the sequence { satisfies an MDP with speed n
−1 2
an and good rate function
I (x) = sup {λx − Λ(λ)} = λ∈R
x2 2
,
∀ x ∈ R.
log dn −n an
, n ≥ 1}
106
L. Fang / Statistics and Probability Letters 98 (2015) 98–106
Acknowledgments We thank Dr. Rui Kuang and Dr. Yuanyuan Ren for their useful discussions. We would also like to thank the anonymous referee for the detailed and helpful comments and suggestions. References Dembo, A., Zeitouni, O., 1998. Large Deviations Techniques and Applications, second ed. Springer, New York. Erdős, P., Rényi, A., Szüsz, P., 1958. On Engel’s and Sylvester’s series. Ann. Univ. Sci. Budapest. Eötvös. Sect. Math. 1, 7–32. Fan, A., Wang, B., Wu, J., 2007. Arithmetic and metric properties of Oppenheim continued fraction expansions. J. Number Theory 127, 64–82. Hartono, Y., Kraaikamp, C., Schweiger, F., 2002. Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients. J. Théor. Nombres Bordeaux 14, 497–516. Hu, W., 2015. Moderate deviation principles for Engel’s, Sylvester’s series and Cantor’s products. Statist. Probab. Lett. 96, 247–254. Mehrdad, B., Zhu, L., 2013. Moderate and large deviations for Erdős–Kac theorem. Arxiv Preprint. arXiv:1311.6180. Rényi, A., 1962. A new approach to the theory of Engel’s series. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 5, 25–32. Touchette, H., 2009. The large deviation approach to statistical mechanics. Phys. Rep. 478, 1–69. Varadhan, S.R.S., 1984. Large Deviations and Applications. SIAM, Philadelphia. Zhu, L., 2014. On the large deviations for Engel’s, Sylvester’s series and Cantor’s products. Electron. Comm. Probab. 19, 1–9.