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

Dingcheng Wang ; Chun Su ; Zhishui Hu

Dalʹnevostočnyj matematičeskij žurnal, Tome 3 (2002) no. 1, pp. 34-51

Résumé

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 $\{S(t)\}_{t \ge 0}$.