A non-clustering property of stationary sequences. (English) Zbl 0536.60044

Summary: For a random sequence of events, with indicator variables \(X_ i\), the behavior of the expectation \(E\{(X_ k+...+X_{k+m-1})/(X_ 1+...+X_ n)\}\) 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.


60G10 Stationary stochastic processes
26D15 Inequalities for sums, series and integrals
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