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 AB 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 XA. 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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Received: 8 June 2001 / Revised version: 30 October 2002 Published online: 15 April 2003
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ID="*" Supported by NSF Grant DMS-99-72417.
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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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DOI: https://doi.org/10.1007/s00440-002-0252-0