Open Access
February, 1981 Limiting Behavior of a Process of Runs
B. G. Pittel
Ann. Probab. 9(1): 119-129 (February, 1981). DOI: 10.1214/aop/1176994512


Let $X_1, X_2, \cdots$ be independent identically distributed (i.i.d.) random variables with a continuous distribution function. Let $R_0 = 0, R_k = \min \{j: j > R_{k - 1}$ and $X_j > X_{j + 1}\}$ and $T_k = R_k - R_{k - 1}, k \geq 1$. We prove that a process $T^{(n)} = \{T_{k + n}\}^\infty_{k = 1}$ converges, in the sense of distribution functions, exponentially fast to a strongly mixing ergodic process. It is shown that $(\max_{1 \leq k \leq n} T_k)/\log n(\log \log n)^{-1} \rightarrow 1$ almost surely and in $L_p, p > 0$. Also, the number of runs $T_k, 1 \leq k \leq n$, larger than or equal to some $m$ is proven to be Poisson distributed in the limit, if $n/m$! converges to a positive number.


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B. G. Pittel. "Limiting Behavior of a Process of Runs." Ann. Probab. 9 (1) 119 - 129, February, 1981.


Published: February, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0453.60017
MathSciNet: MR606801
Digital Object Identifier: 10.1214/aop/1176994512

Primary: 60C05
Secondary: 05A15 , 60F05

Keywords: convergence to Poisson process , ergodic process , Independent identically distributed random variables , MacMahon formula , mixing property , Runs , the longest run distribution

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 1 • February, 1981
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