A Non-Clustering Property of Stationary Sequences

Abstract

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 $k$.