An optimal bound on the tail distribution of the number of recurrences of an event in product spaces
(2003) In Probability Theory and Related Fields19850101+01:00 126(1). p.5160 Abstract
 Let X1, X2,... be independent random variables and a a positive real number. For the sake of illustration, suppose A is the event that Xi+1 +...+ Xj 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 upperbound 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 = (X1, X2,...) 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 X1, X2,... be independent random variables and a a positive real number. For the sake of illustration, suppose A is the event that Xi+1 +...+ Xj 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 upperbound 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 = (X1, X2,...) 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 finitedimensional 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,..., Xj) 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(Nln(P(Ac)) greater than or equal to 1), where Ngamma 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(Nln(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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/308356
 author
 Klass, MJ and Nowicki, Krzysztof ^{LU}
 organization
 publishing date
 2003
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 number of entrance times, number of event recurrences, bounds, Poisson, tail probability inequalities, HoffmannJorgensen inequality, product, spaces
 in
 Probability Theory and Related Fields19850101+01:00
 volume
 126
 issue
 1
 pages
 51  60
 publisher
 Springer
 external identifiers

 wos:000183544800003
 scopus:0038047432
 ISSN
 01788051
 DOI
 10.1007/s0044000202520
 language
 English
 LU publication?
 yes
 id
 6aa555805b02499e909852e53c914b43 (old id 308356)
 date added to LUP
 20070803 12:16:59
 date last changed
 20180529 11:24:38
@article{6aa555805b02499e909852e53c914b43, abstract = {Let X1, X2,... be independent random variables and a a positive real number. For the sake of illustration, suppose A is the event that Xi+1 +...+ Xj 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 upperbound 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 = (X1, X2,...) 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 finitedimensional 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,..., Xj) 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(Nln(P(Ac)) greater than or equal to 1), where Ngamma 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(Nln(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 = {01788051}, keyword = {number of entrance times,number of event recurrences,bounds,Poisson,tail probability inequalities,HoffmannJorgensen inequality,product,spaces}, language = {eng}, number = {1}, pages = {5160}, publisher = {Springer}, series = {Probability Theory and Related Fields19850101+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/10.1007/s0044000202520}, volume = {126}, year = {2003}, }