Large deviations for random walks under subexponentiality: The big-jump domain

D. Denisov, A. B. Dieker, V. Shneer

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    Abstract

    For a given one-dimensional random, walk {Sn} with a subexponential step-size distribution, we present a unifying theory to study the sequences {xn} for which P{Sn > x} ∼ nP{S 1 x} as n → ∞ uniformly for x ≥ xn. We also investigate the stronger "local" analogue, P{Sn ∈ (x, x + T]} ∈ nP{S1 ∈ (x, x + T]}. Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory. When specialized to the most important subclasses of subexponential distributions that have been studied in the literature, we reproduce known theorems and we supplement them with new results. © 2008 Institute of Mathematical Statistics.
    Original languageEnglish
    Pages (from-to)1946-1991
    Number of pages45
    JournalAnnals of Probability
    Volume36
    Issue number5
    DOIs
    Publication statusPublished - Sep 2008

    Keywords

    • Large deviations
    • Random walk
    • Subexponentiality

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