Asymptotics of randomly stopped sums in the presence of heavy tails

By Denis Denisov, Serguei Foss, Dmitry Korshunov. 2008

83 citations recorded by the source corpus.

We study conditions under which P{S� > x} ∼ P{M� > x} ∼ EP{�1 > x} as x → ∞, where Sis a sum �1 + ... + �� of random sizeand Mis a maximum of partial sums M� = maxn�� Sn. Heren, n = 1, 2, . . . , are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case whereis independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where E� > 0 and where the tail ofis comparable with or heavier than that of �, and obtain the asymptotics P{S� > x} ∼ EP{�1 > x} + P{� > x/E�} as x → ∞. This case is of a primary interest in the branching processes. In addition, we obtain new uniform (in all x and n) upper bounds for the ratio P{Sn > x}/P{�1 > x} which substantially improve Kesten's bound in the subclass Sof subexpo-

Subfields: Management Science and Operations Research, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/25699648939c423bcfc53f93385795f7c1966667

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In Management Science and Operations Research

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Asymptotics for Weighted Random Sums (2011, background): “Moreover, the results in [9] also include the case where the asymptotic behavior of the randomly stopped sum is not solely determined by the one-big-jump principle, and, in particular, it was shown that”
The Strong Law of Large Numbers for Extended Negatively Dependent Random Variables (2010, background): “See also Aleškevičienė et al. (2008) and Denisov et al. (2010) for some extensions.”