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Singularity analysis for heavy-tailed random variables | | Nicholas M. Ercolani
; Sabine Jansen
; Daniel Ueltschi
; | | Date: |
17 Sep 2015 | | Abstract: | Let $alpha in (0,1),c>0$ and $G(z) = csum_{k=1}^infty z^k exp( -
k^alpha)$. We apply complex analysis to determine the asymptotic behavior of
the $m$th coefficient of the $n$th power of $G(z)$ when $m,n o infty$ with $m
geq n G’(1)/G(1)$ and recover five theorems by A. V. Nagaev (1968) on sums of
i.i.d. random variables with stretched exponential law $mathbb{P}(X_1=k) = c
exp(- k ^alpha)$. From the point of view of complex analysis, the main
novelty is the combination of singularity analysis, Lindel"of integrals, and
bivariate saddle points. From the point of view of probability, the proof
provides a new, unified approach to large and moderate deviations for
heavy-tailed, integer-valued random variables. | | Source: | arXiv, 1509.5199 | | Services: | Forum | Review | PDF | Favorites | |
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