An optimal bound on the tail distribution of the number of recurrences of an event in product spaces
Author
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.
Department/s
Publishing year
2003
Language
English
Pages
51-60
Publication/Series
Probability Theory and Related Fields
Volume
126
Issue
1
Document type
Journal article
Publisher
Springer
Topic
- Probability Theory and Statistics
Keywords
- number of entrance times
- number of event recurrences
- bounds
- Poisson
- tail probability inequalities
- Hoffmann-Jorgensen inequality
- product
- spaces
Status
Published
ISBN/ISSN/Other
- ISSN: 0178-8051