On the length of the longest head run

Statistics and Probability Letters 130 (2017) 111–114

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On the length of the longest head run S.Y. Novak MDX University, School of Science & Technology, The Burroughs, London NW44BT, UK

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Article history: Received 24 December 2016 Received in revised form 20 June 2017 Accepted 21 June 2017 Available online 4 August 2017

a b s t r a c t We evaluate the accuracy of approximation to the distribution of the length of the longest head run in a Markov chain with a discrete state space. An estimate of the accuracy of approximation in terms of the total variation distance is established for the first time. © 2017 Published by Elsevier B.V.

MSC: 60E15 60G70 Keywords: Longest head run Extremes in samples of random size

1. Introduction Let {ξi , i ≥ 1} be a sequence of random 0’s and 1’s (i.e., ‘‘tails’’ and ‘‘heads’’). Then Ln = max{k : ξi+1 = · · · = ξi+k = 1 (∃i ≤ n − k)}

(1)

is the length of the longest head run (LLHR) among ξ1 , . . . , ξn . Statistic Ln has applications in biology, reliability theory, finance, and nonparametric statistics (see, e.g., Balakrishnan and Koutras, 2001; Barbour et al., 1992; Bateman, 1948; Schwager, 1983). In particular, the reliability of a consecutive k-out-of-n system with n components can be expressed via P(Ln < k), where the event {ξk = 1} represents failure of the kth component: the system fails if and only if k consecutive components fail (Balakrishnan and Koutras, 2001; Chryssaphinou and Papastavridis, 1990; Fu et al., 2003; Novak, 1989). The study of the distribution of LLHR has a long history. Apparently, the task was first formulated by de Moivre (1738), Problem LXXIV. Renewed interest to the topic arose in connection with the Erdös–Rényi strong law of large numbers (Erdös and Rényi, 1970). A limit theorem for LLHR in the case of independent Bernoulli B(p) trials was established by Goncharov (1944). The limiting distribution of LLHR was found in more general situations as well, see Balakrishnan and Koutras (2001), Novak (1988), Novak (1991) and Vaggelatou (2003) and references therein. In particular, a limit theorem for LLHR in a Markov chain with a finite state space X where hitting a subset of X is considered a ‘‘success’’ is given in Novak (1988). An estimate of the rate of convergence and asymptotic expansions in the limit theorem for LLHR in a two-state Markov chain have been established in Novak (1989). Concerning LIL-type results, see Novak (2011) and references therein. An exact formula for P(Ln < k) in terms of combinatorial coefficients in the case of independent Bernoulli trials was found by Uspensky (1937). In the case of a two-state Markov chain Fu et al. (2003) present an exact formula for P(Ln < k) in terms of a specially constructed matrix of transition probabilities, and establish the asymptotics of ln P(Ln < k) as n → ∞ if k is fixed (see also Theorem 2 in Novak, 1989). E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2017.06.020 0167-7152/© 2017 Published by Elsevier B.V.

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Note that Ln can be represented as a sample maximum in a sample of random size νn , where νn is a certain renewal process (cf. Novak, 1989, 1991). References concerning extremes in samples of random size can be found, e.g., in Galambos (1987), Lebedev (2015) and Novak (2011). It is known that the accuracy of approximation to the distribution of LLHR in terms of the uniform distance is n−1 ln n (Novak, 1989). The result has been generalised to the case of a Markov chain with a finite state space (Novak, 1991) as well as to the case of m-dependent random variables (Novak, 1992). Asymptotic expansions in the limit theorem for LLHR in a two-state Markov chain (Novak, 1989) confirm that the rate n−1 ln n cannot be improved. There is a simple relation between LLHR and the number Nn (k) of head runs with lengths ≥ k:

{Ln < k} = {Nn (k) = 0}. Note that the estimates of the accuracy of approximation to the distribution of Nn (k) have been established in terms of the total variation distance (see Balakrishnan and Koutras, 2001; Barbour et al., 1992; Novak, 2011 and references therein). However, the problem of evaluating the accuracy of approximation to the distribution of LLHR in terms of the total variation distance remained open for a long while. In this paper we derive an estimate of order n−1 ln n to the total variation distance between L(Ln ) and the approximating distribution. 2. Results Let {Xi , i ≥ 1} be a homogeneous Markov chain with a finite state space X and transition probabilities ∥pij ∥i,j∈X . We denote by

π¯ = ∥πi ∥i∈X the stationary distribution of the chain. Given a subset A ⊂ X , let LLHR be defined by (1), where

ξi = 1{Xi ∈ A} (hitting A is considered a ‘‘success’’). We set U = ∥pij ∥i,j∈A , π¯ A = ∥πi ∥i∈A , and let q(k) = π¯ A U k−1 (E − U)1¯

(k ≥ 1),

where 1¯ is a vector of 1’s and E is a unit diagonal matrix. Let ζn , Zn be random variables (r.v.s) with distribution functions (d.f.s) P(ζn < k) = (1 − q(k))n−k , P(Zn < k) = exp(−nq(k))

(k ≥ 1).

Recall the definition of the total variation distance between the distributions of r.v.s X and Y : dTV (X ; Y ) ≡ dTV (L(X ); L(Y )) = sup |P(X ∈ A) − P(Y ∈ A)| , A∈ A

where A is a Borel σ -field. The distribution of LLHR Ln can be well approximated by L(ζn ) or L(Zn ); the accuracy of such approximation in terms of the uniform distance is known to be of order n−1 ln n. In Theorem 1 we show that the result holds in terms of the stronger total variation distance. Theorem 1. Assume that (P0) there is only one class C of essential states that consists of periodic sub-classes C1 , . . . , Cd ; (P1) A ∩ Ci ̸ = Ø (1 ≤ i ≤ d); (P2) 0 < λ < 1, where λ is the largest eigenvalue of matrix U; (P3) if i ∈ Cℓ for some ℓ ∈ {1, . . . , d}, then

|pij (m) − dπj | ≤ um , H :=

∑

um < ∞

(2)

m≥1

if j ∈ A ∩ Ck and k − m = ℓ (mod d); if i ∈ / C1 ∪ · · · ∪ Cd , then (2) holds for all j ∈ A; (P4) zi > 0 (∀i ∈ A), where z¯ = ∥zi ∥i∈A is the corresponding to λ right eigenvector of matrix U. Then there exists a positive constant C = C (λ, z¯ , π¯ A ) such that dTV (Ln ; Zn ) ≤ Cn−1 ln n

(n ≥ C ).

The result holds if Zn in (3) is replaced with ζn .

(3)

S.Y. Novak / Statistics and Probability Letters 130 (2017) 111–114

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3. Proofs Proof of Theorem 1 makes use of Theorem 2 from Novak (1991), which is presented below (note that the argument of Theorem 2 in Novak (1991) is valid for any fixed d ∈ N). In the particular case of a stationary Markov chain the result of Theorem 2 is given by Theorem 2.1 in Novak (1988). Theorem 2. Let {Xi , i ≥ 1} be a homogeneous Markov chain with a discrete state space X , transition probabilities ∥pij ∥i,j∈X and stationary distribution π. ¯ Assume conditions (P0)–(P4). Then there exists a positive constant c⋆ = c⋆ (λ, z¯ , π¯ A ) such that as n > 2k ≥ c⋆ ,

( ) |P(Ln < k) − P(Zn < k)| ≤ c⋆ λk + c⋆ kλk exp −nq(k)(1 − c⋆ kλk ) .

(4)

Taking into account the obvious inequality (x, y ∈ R),

|ex − ey | ≤ |x − y|emax{x;y}

(5)

we notice that (4) holds true if Zn is replaced with ζn . In the case of independent observations inequalities of this kind with explicit constants are presented in Novak (1992) and Muselli (2000). In the case of a two-state Markov chain with α := p11 ∈ (0; 1), β := p00 < 1, a sharp bound of this kind is given in Novak (1989, Theorem 2): there exist constants q ∈ (0; 1), C < ∞ such that sup ⏐P(Ln < k) − A(t0 )/t0n+1 ⏐ ≤ Cqn

⏐

⏐

(6)

k>C

for particular t0 and function A(t) obeying |A(t0 ) − 1| ≤ C1 γ kα k , |t0 − 1 − γ α k | ≤ C1 k(γ α k )2 for some C1 < ∞, where γ = (1 − α )(1 − β )/α (2 − α − β ). In the case of independent Bernoulli B(α ) trials (6) holds with q = α, C = (2 + α − α 2 )/ (1 − α )(1 − α 2 ). By a well-known property of the total variation distance, 2dTV (Ln ; Zn ) =

∑ |P(Ln = k) − P(Zn = k)|.

(7)

k≥0

The idea of the prof is to split the sum in (7) into appropriate fragments and show that the desired estimate holds for each fragment. Recall that π¯ A = ∥πi ∥i∈A , and set c∗ = ⟨π¯ A ; z¯ ⟩(1 − λ)/λz ∗ , c ∗ = ⟨π¯ A ; z¯ ⟩(1 − λ)/λz∗ , z¯∗ = inf{zj : j ∈ A} ,

z¯ ∗ = sup{zj : j ∈ A}.

Note that 0 < c∗ ≤ c ∗ < ∞. It is easy to see that c∗ λk ≤ q(k) ≤ c ∗ λk

(8)

(cf. (8) in Novak, 1991). Let k(n) = log n − log ln n + log(c∗ /2) . Hereinafter log is to the base 1/λ, symbol c (with or without indexes) denotes positive constants. Using (4) and (8), we check that

∑

|P(Ln = k) − P(Zn = k)|

k≤k(n)

≤ P(Ln ≤ k(n)) + P(Zn ≤ k(n)) ≤ c1 n−1 ln n.

(9)

It remains to evaluate

∑

|P(Ln = k) − P(Zn = k)|.

k>k(n)

According to (4) and (8), there exists a positive constant c2 such that

|P(Ln = k) − P(Zn = k)| ≤ c2 λk + c2 kλk e−nλ

∗ /2

kc

(10)

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S.Y. Novak / Statistics and Probability Letters 130 (2017) 111–114

as n > 2k ≥ c2 . Evidently,

∑

λk ≤ λk(n) /(1 − λ) = 2n−1 (ln n)/(1 − λ)c∗ .

(11)

k>k(n)

Thus, it remains to evaluate k>k(n) kλk e−nλ c∗ /2 . Note that function f (x) = xe−x decreases in [1; ∞). Clearly, nλk c∗ /2 ∈ [1; ln n] as k(n) < k < log(nc∗ /2) . Therefore, k

∑

kλk e−nλ

∑

kc

∗ /2

k(n)
nλk e−nλ

∑

≤ n−1 log(nc∗ /2)

kc

∗ /2

k(n)
≤ n−1 log(nc∗ /2)

∫

log(nc∗ /2)

nλx e−nλ

xc

∗ /2

dx

k(n)

≤ 2n−1 log(nc∗ /2)/ln(1/λ)c∗ .

(12)

Since

∑

kλk ≤ mλm /(1 − λ)2

(m ≥ 1),

k≥m

we have

∑

kλk e−nλ

k≥log(nc∗ /2)

kc

∗ /2

≤

∑

kλk ≤ 2⌈log(nc∗ /2)⌉/nc∗ (1 − λ)2 ,

(13)

k≥log(nc∗ /2)

where ⌈x⌉ denotes the smallest integer greater than or equal to x. Combining estimates (9)–(13), we derive (3). The proof is complete. □ Acknowledgments The author is grateful to the Associate Editor and the reviewers for helpful comments. References Balakrishnan, N., Koutras, M.V., 2001. Runs and Scans with Applications. Wiley, New York. Barbour, A.D., Holst, L., Janson, S., 1992. Poisson Approximation. Clarendon Press, Oxford. Bateman, G.I., 1948. On the power function of the longest run as a test for randomness in a sequence of alternatives. Biometrika 35, 97–112. Chryssaphinou, O., Papastavridis, S.G., 1990. Limit distribution for a consecutive-k-out-of-n:F system. Adv. Appl. Prob. 22, 491–493. de Moivre, A., 1738. The Doctrine of Chances. H Woodfall, London. Erdös, P., Rényi, A., 1970. On a new law of large numbers. J. Anal. Math. 22, 103–111. Fu, J.C., Wang, L., Lou, W.Y.W., 2003. On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials. J. Appl. Probab. 40 (2), 346–360. Galambos, J., 1987. The Asymptotic Theory of Extreme Order Statistics. RE Krieger Publishing Co, Melbourne. Goncharov, V.L., 1944. On the field of combinatory analysis. Amer. Math. Soc. Transl. 19 (2), 1–46. Lebedev, A.V., 2015. Non-Classical Problems in Extreme Value Theory. (D.Sc. thesis), Moscow State University, Moscow. Muselli, M., 2000. New improved bounds for reliability of consecutive-k-out-of-n:F systems. J. Appl. Probab. 37, 1164–1170. Novak, S.Y., 1988. Time intervals of constant sojourn of a homogeneous Markov chain in a fixed subset of states. Siberian Math. J. 29 (1), 100–109. Novak, S.Y., 1989. A symptotic expansions in the problem of the longest head–run for a Markov chain with two states. Trudy Inst. Math. 13, 136–147 (in Russian). Novak, S.Y., 1991. Rate of convergence in the limit theorem for the length of the longest head run. Siberian Math. J. 32 (3), 444–448. Novak, S.Y., 1992. Longest runs in a sequence of m–dependent random variables. Probab. Theory Related Fields 91, 269–281. Novak, S.Y., 2011. Extreme Value Methods with Applications to Finance. Chapman & Hall/CRC Press, London, ISBN: 9781439835746. Schwager, S.J., 1983. Run probabilities in sequences of Markov-dependent trials. J. Amer Stat. Assoc. 78 (1), 168–175. Uspensky, J.V., 1937. Introduction to Mathematical Probability. McGraw-Hill Book Company. Vaggelatou, E., 2003. On the length of the longest run in a multi-state Markov chain. Statistics Probab. Letters 62, 211–221.