Far Eastern Mathematical Journal

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Precise large deviation for random sums of random walks with dependent heavy-tailed steps


Dingcheng Wang, Chun Su, Zhishui Hu

2002, issue 1, Ñ. 34–51


Abstract
In most applications the assumption of independent step sizes is, clearly, unrealistic. It is an important way to model the dependent steps $\{X_n \}_{n=1}^{\infty}$ of the random walk as a two-sided linear process, $X_n=\sum\limits_{j=-\infty}^{\infty}\varphi_{n-j} \eta_j$, $n=1,2,3,\dots$, where $\{\eta,\eta_n,\ n=0,\pm 1,\pm 2,\pm 3,\dots\}$ is a sequence of iid random variables with finite mean $\mu>0$. Moreover suppose that $\eta$ satisfies certain tailed balance condition and its distribution function belongs to $ERV(-\alpha,-\beta)$ with $1<\alpha\le\beta<\infty$. Denote $S_n=X_1+X_2+\dots+X_n$, $n\ge 1$. At first we discuss precise large deviation problems of non-random sums $\{S_n-ES_n\}_{ n=1}^{\infty}$, then discuss precise large deviation problems of $S(t)-ES(t)=\sum_{i=1}^{N(t)}(X_i-EX_i)$, $t\ge 0$ for non-negative and inter-value random process $N(t)$ such that Assumption A, independent of $\{\eta_n\}_{n=-\infty}^{\infty}$. We show that if the steps of random walk are not independent, then precise large deviation result of random sums may be different from the case with iid steps, which means that dependence affects the tails of compound processes $\{S(t)\}_{t \ge 0}$.

Keywords:
Class ERV Dependent, Heavy-tailed Distribution, Random Walk, Precise Large Deviation, Tail Balance Condition, Two-sided linear process.

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