Dmitry Korshunov

An introduction to heavy-tailed and subexponential distributions

Preface.- Introduction.- Heavy- and long-tailed distributions.- Subexponential distributions.- Densities and local probabilities.- Maximum of random walks.- References.- Index

Asymptotics for sums of random variables with local subexponential behaviour

We study distributions F on [0,∞) such that for some T ≤ ∞, F*2(x, x+T] ∼ 2F(x, x+T]. The case T = ∞ corresponds to F being subexponential, and our analysis shows that the properties for T < ∞ are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman–Harris branching processes.

Asymptotics of randomly stopped sums in the presence of heavy tails

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 Kestens bound in the subclass Sof subexpo-

On lower limits and equivalences for distribution tails of randomly stopped sums

For a distribution F*τ of a random sum Sτ=ξ1+⋯+ξτ of i.i.d. random variables with a common distribution F on the half-line [0, ∞), we study the limits of the ratios of tails as x→∞ (here, τ is a counting random variable which does not depend on {ξn}n≥1). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.

On Large Delays in Multi-Server Queues with Heavy Tails

We present upper and lower bounds for the tail distribution of the stationary waiting time D in the stable GI/GI/s first-come first-served (FCFS) queue. These bounds depend on the value of the traffic load ρ which is the ratio of mean service and mean interarrival times. For service times with intermediate regularly varying tail distribution the bounds are exact up to a constant, and we are able to establish a “principle of s-k big jumps” in this case (here k is the integer part of ρ), which gives the most probable way for the stationary waiting time to be large. Another corollary of the bounds obtained is to provide a new proof of necessity and sufficiency of conditions for the existence of moments of the stationary waiting time.

Lower limits for distribution tails of randomly stopped sums

We study lower limits for the ratio

On the asymptotic Laplace method and its application to random chaos

\overline{F^{*\tau}}(x)/\,\overline F(x)

How to measure the accuracy of the subexponential approximation for the stationary single server queue

of tail distributions, where

Heavy-Tailed and Long-Tailed Distributions

F^{*\tau}

Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case

is a distribution of a sum of a random size