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February, 1984 A Non-Clustering Property of Stationary Sequences
Arif Zaman
Ann. Probab. 12(1): 193-203 (February, 1984). DOI: 10.1214/aop/1176993382


For a random sequence of events, with indicator variables $X_i$, the behavior of the expectation $E\{(X_k + \cdots + X_{k + m - 1})/(X_1 + \cdots + X_n)\}$ for $1 \leq k \leq k + m - 1 \leq n$ can be taken as a measure of clustering of the events. When the measure on the $X$'s is i.i.d., or even exchangeable, a symmetry argument shows that the expectation can be no more than $m/n$. When the $X$'s are constrained only to be a stationary sequence, the bound deteriorates, and depends on $k$ as well. When $m/n$ is small, the bound is roughly $2m/n$ for $k$ near $n/2$ and is like $(m/n) \log n$ for $k$ near 1 or $n$. The proof given is partly constructive, so these bounds are nearly achieved, even though there is room for improvement for other values of $k$.


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Arif Zaman. "A Non-Clustering Property of Stationary Sequences." Ann. Probab. 12 (1) 193 - 203, February, 1984.


Published: February, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0536.60044
MathSciNet: MR723738
Digital Object Identifier: 10.1214/aop/1176993382

Primary: 60G10
Secondary: 26D15

Keywords: clustering , cyclic sums , Stationary sequences

Rights: Copyright © 1984 Institute of Mathematical Statistics

Vol.12 • No. 1 • February, 1984
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