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

Zaman, Arif

Ann. Probab. 12, 193-203 (1984).

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.

MSC:

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

Keywords:

cyclic sums; exchangeable; stationary sequence