On the Lower Tail Probabilities of Some Random Series M.A. Lifshits The behavior ofP tail probabilities P {S ≤ r}, r → 0, is investigated, where S is a series S = λj Zj generated by some sequence of positive numbers {λj } and by a sequence {Zj } of independent copies of a positive random variable Z. We present the exact asymptotic expression for P {S ≤ r} by means of Laplace transform L(γ) = E exp{−γS} under weak assumptions on the behavior of the tail probabilities of Z in the vicinity of zero. The bounds of accuracy are also given, and under weak supplementary smoothness conditions the asymptotic properties of the density of S are investigated.
On the Lower Tail Probabilities of Some Random Series M.A. Lifshits
M.A. Lifshits. The behavior of tail probabilities P{S ⤠r}, r â 0, is investigated, where S is a series S = â λjZj generated by some sequence of positive numbers ...
Integration over white noise. Spectral representation of a ... Transformation of the random function into continuous one by the change of metric. Connection ...
E Xk = 0 and EX2 k = Ï2 k for all k. Hoeffding 1963, Theorem 3, proved that. P{Mn ⥠nt} ⤠Hn(t, p), H(t, p) = `1 + qt/p´ p+qt`1 â t´qâqt with q = 1. 1 + Ï2 , p = 1 â q, ...
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where d and Ï are some important numeric parameters of the field x and Φ is the distribution function of the standard normal law. We apply this relation to the investigation of the Gaussian measure of large balls in the space lp in order to general
The typical problem of the small ball probabilities is to investigate the asymptotic behavior of P{||X|| < r}, r â 0, for a random vector X taking values in a normed space. As a generic example, take a sample path of a ran- dom function as X and co
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On the Supremum of Random Dirichlet Polynomials. Mikhail Lifshits and Michel Weber. We study the supremum of some random Dirichlet polynomials. DN (t) =.
Our statistics of interest are the maximum of a random field G, resulting from the .... by a postdoctoral fellowship from the ISM-CRM, Montreal, Quebec, Canada. 1.
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