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

Klass, MJ and Nowicki, Krzysztof LU (2003) In Probability Theory and Related Fields1985-01-01+01:00 126(1). p.51-60
Abstract
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... (More)
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. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
number of entrance times, number of event recurrences, bounds, Poisson, tail probability inequalities, Hoffmann-Jorgensen inequality, product, spaces
in
Probability Theory and Related Fields1985-01-01+01:00
volume
126
issue
1
pages
51 - 60
publisher
Springer
external identifiers
  • wos:000183544800003
  • scopus:0038047432
ISSN
0178-8051
DOI
language
English
LU publication?
yes
id
6aa55580-5b02-499e-9098-52e53c914b43 (old id 308356)
date added to LUP
2007-08-03 12:16:59
date last changed
2018-05-29 11:24:38
@article{6aa55580-5b02-499e-9098-52e53c914b43,
  abstract     = {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 &lt; j &lt; 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 &gt; 0. Without further assumptions we prove that, if 0 &lt; P (A) &lt; 1, then P (L(A) greater than or equal to k) &lt; 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.},
  author       = {Klass, MJ and Nowicki, Krzysztof},
  issn         = {0178-8051},
  keyword      = {number of entrance times,number of event recurrences,bounds,Poisson,tail probability inequalities,Hoffmann-Jorgensen inequality,product,spaces},
  language     = {eng},
  number       = {1},
  pages        = {51--60},
  publisher    = {Springer},
  series       = {Probability Theory and Related Fields1985-01-01+01:00},
  title        = {An optimal bound on the tail distribution of the number of recurrences of an event in product spaces},
  url          = {http://dx.doi.org/},
  volume       = {126},
  year         = {2003},
}