Random walk with a heavy-tailed jump distribution

The classical random walk of which the one-step displacement variable $mathbf x$ has a first finite negative moment is considered. The R.W. possesses an unique stationary distribution; $mathbf x$ is a random variable with this distribution. It is assumed that the righthand and/or the lefthand tail of the distribution of ${mathbf u$ are heavy-tailed. For the type of heavy-tailed distribution considered in this study a contraction factor ${Delta (a)$ exists with ${Delta (a) {downarrow 0$ for $a uparrow 1$, and $a uparrow 1$ is equivalent with $E { {mathbf u {uparrow 0$. It is shown that ${Delta (a) {mathbf x$ converges in distribution for $a uparrow 1$. It is the analysis of the tail of this limiting distribution of ${Delta (a) {mathbf x$ which is the main purpose of the present study in particular when ${mathbf u$ is a mix of stochastic variables ${mathbf u_i , i = 1 , ldots , N$, each ${mathbf u_i$ having its own heavy tail characteristics for its right- and lefthand tails. For an important case it is shown that for the tail of the distribution of ${Delta (a) {mathbf x$ an asymptotic expression in the variables ${Delta (a)$ and $t$ for ${Delta (a) {downarrow 0$ and ${t rightarrow infty$ can be derived. For this asymptotic relation the dominating term is completely determined by the heavier tail of the $2N$ tails of the ${mathbf u_i$; the other terms of the asymptotic relation show the influence of less heavier tails and, depending on $t$, the terms may have a contribution which is not always negligible. The study starts with the derivation of a functional equation for the L.S.-transform of the distribution of ${mathbf x$ and that of the excess distribution of the stationary idle time distribution. For several important cases this functional equation could be solved and thus has led to the above mentioned asymptotic result. The derivation of it required quite some preparation, because it needed an effective description of the heavy-tailed jump vector ${mathbf u$. It was obtained by prescribing the heavy-tailed distributions of $[{mathbf u]^+ = max (0 , {mathbf u )$ and $[{mathbf u]^- = min (0, {mathbf u)$. The random walk may serve as a model for the actual waiting process of a GI/G/1 queueing model; in that case the distribution of ${mathbf u$ is that of the difference of the service time and the interarrival time. The analysis of the present study then describes the heavy-traffic theory for the case with heavy-tailed service-and/or interarrival time distribution.