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An optimal bound on the tail distribution of the number of recurrences of an event in product spaces
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  • Published: May 2003

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

  • Michael J. Klass1 &
  • Krzysztof Nowicki2 

Probability Theory and Related Fields volume 126, pages 51–60 (2003)Cite this article

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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 |X i+1 +...+X j |≥a for some integers 0≤i<j<∞. For each k≥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 ,...)Ω=Π j ≥1Ω j be a random element in a product probability space (Ω,ℬ,P=⊗ j ≥1 P j ). We are interested in events A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] =(X i+1 ,...,X j ) ensures that XA. By definition, for sequentially searchable A, P(A)≡P(L(A)≥1)=P(𝒩−ln (P(Ac)) ≥1), where 𝒩γ denotes a Poisson random variable with some parameter γ>0. Without further assumptions we prove that, if 0<P(A)<1, then P(L(A)≥k)<P(𝒩−ln (P(Ac)) ≥k) for all integers k≥2.

An application to sums of independent Banach space random elements in l ∞ is given showing how to extend our theorem to situations having dependent components.

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Authors and Affiliations

  1. University of California, Departments of Mathematics and Statistics, Berkeley, CA 94720-3840, USA, , , , , , US

    Michael J. Klass

  2. Lund University, Department of Statistics, Box 743, S-220 07 Lund, Sweden. e-mail: Krzysztof.Nowicki@stat.lu.se, , , , , , SE

    Krzysztof Nowicki

Authors
  1. Michael J. Klass
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  2. Krzysztof Nowicki
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Additional information

Received: 8 June 2001 / Revised version: 30 October 2002 Published online: 15 April 2003

RID="*"

ID="*" Supported by NSF Grant DMS-99-72417.

RID="†"

ID="†" Supported by the Swedish Research Council.

Mathematics Subject Classification (2000): Primary 60E15, 60G50

Key words or phrases: Tail probability inequalities – Hoffmann-Jo rgensen inequality – Poisson bounds – Number of event recurrences – Number of entrance times – Product spaces

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Klass, M., Nowicki, K. An optimal bound on the tail distribution of the number of recurrences of an event in product spaces. Probab. Theory Relat. Fields 126, 51–60 (2003). https://doi.org/10.1007/s00440-002-0252-0

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  • Issue Date: May 2003

  • DOI: https://doi.org/10.1007/s00440-002-0252-0

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Keywords

  • Banach Space
  • Real Number
  • Probability Space
  • Positive Real Number
  • Product Space
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