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An optimal bound on the tail distribution of the number of recurrences of an event in product spaces

In English

Författare

Summary, in English

Let X-1, X-2,... be independent random variables and a a positive real number. For the sake of illustration, suppose A is the event that Xi+1 +...+ X-j greater than or equal to a for some integers 0 less than or equal to i < j < infinity. For each k greater than or equal to 2 we upper-bound the probability that A occurs k or more times, i.e. that A occurs on k or more disjoint intervals, in terms of P(A), the probability that A occurs at least once. More generally, let X = (X-1, X-2,...) is an element of Omega = Pi(jgreater than or equal to1) Omega(j) be a random element in a product probability space (Omega, B, P = circle times(jgreater than or equal to1)P(j)). We are interested in events A is an element of B that are (at most contable) unions of finite-dimensional cylinders. We term such sets sequentially searchable. Let L(A) denote the (random) number of disjoint intervals (i, j] such that the value of X(i,j] = (Xi+1,..., X-j) ensures that X is an element of A. By definition, for sequentially searchable A, P(A) P(L(A) greater than or equal to 1) = P(N-ln(P(Ac)) greater than or equal to 1), where N-gamma denotes a Poisson random variable with some parameter gamma > 0. Without further assumptions we prove that, if 0 < P (A) < 1, then P (L(A) greater than or equal to k) < P(N-ln(P(Ac)) greater than or equal to k) for all integers k greater than or equal to 2. An application to sums of independent Banach space random elements in l(infinity) is given showing how to extend our theorem to situations having dependent components.

Publiceringsår

2003

Språk

Engelska

Sidor

51-60

Publikation/Tidskrift/Serie

Probability Theory and Related Fields

Volym

126

Issue

1

Dokumenttyp

Artikel i tidskrift

Förlag

Springer

Ämne

  • Probability Theory and Statistics

Nyckelord

  • number of entrance times
  • number of event recurrences
  • bounds
  • Poisson
  • tail probability inequalities
  • Hoffmann-Jorgensen inequality
  • product
  • spaces

Status

Published

ISBN/ISSN/Övrigt

  • ISSN: 0178-8051