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Large deviations probabilities for random walks in the absence of finite expectations of jumps. (English) Zbl 1028.60021

Let \(X_1, X_2, \ldots \) be i.i.d. random variables with regularly varying tails: \[ P(X_1 >t) =V(t)=t^{-\beta}L(t),\quad P(X_1 <-t)=W(t)=t^{-\alpha}L_W(t), \] where \(\alpha \leq \min(1, \beta),\) and \(L\) and \(L_W\) are slowly varying functions as \(t\to\infty\). Set \(S_n =X_1 +\ldots +X_n\), \(\overline{S}_n = \max_{0\leq k\leq n} S_k\). The author presents the asymptotic behaviour of \(P(S_n >x) \to 0\) and \(P(\overline{S}_n >x) \to 0\) as \(x\to\infty\). He also presents a criterion for \(\overline{S}_{\infty}<\infty\) a.s. and proves that under some additional conditions \(P(\overline{S}_{\infty} >x) \sim cV(x)/W(x).\)

MSC:

60F10 Large deviations
60G50 Sums of independent random variables; random walks
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