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Superlarge deviations for sums of random variables with arithmetical super-exponential distributions

Abstract

Local limit theorems are obtained for superlarge deviations of sums S(n) = ξ(1) + ... + ξ(n) of independent identically distributed random variables having an arithmetical distribution with the right-hand tail decreasing faster that that of a Gaussian law. The distribution of ξ has the form ℙ(ξ = k) = \( e^{ - k^\beta L(k)} \), where β > 2, k ∈ ℤ (ℤ is the set of all integers), and L(t) is a slowly varying function as t → ∞ which satisfies some regularity conditions. These theorems describing an asymptotic behavior of the probabilities ℙ(S(n) = k) as k/n → ∞, complement the results on superlarge deviations in [4, 5].

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Correspondence to A. A. Mogulskiĭ.

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Original Russian Text © A. A. Mogulskiĭ and Ch. Pagma, 2008, published in Matematicheskie Trudy, 2008, Vol. 11, No. 1, pp. 81–112.

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Mogulskiĭ, A.A., Pagma, C. Superlarge deviations for sums of random variables with arithmetical super-exponential distributions. Sib. Adv. Math. 18, 185–208 (2008). https://doi.org/10.3103/S1055134408030048

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Keywords

  • arithmetical super-exponential distribution
  • integro-local and local theorems
  • superlarge deviations
  • deviation function
  • random walk
  • Gaussian approximation
  • Poissonian approximation