The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk
Let \(\xi_1,\xi_2,\dots\) be an i.i.d. sequence of random variables having distribution function (df) \(F\). It is throughout assumed that \(F\) belongs to the class of long-tailed df's, i.e. satisfies \[ \overline F(x)> 0,\;x\in\mathbb{R},\quad \text{and}\quad \lim_{x\to\infty} {\overline F(x- h)\over\overline F(x)}= 1\text{ for all }h> 0.\tag{\(*\)} \] (Here, for any df \(G\) (on \(\mathbb{R}\)) \(\overline G(x):= 1- G(x)\), \(x\in\mathbb{R}\).) Furthermore, it is assumed that the expectation of \(\xi_1\) exists and is zero. The corresponding random walk \((S_n)\) is defined by \(S_0= 0\) and \(S_n= \xi_1+\cdots+ \xi_n\) \((n\geq 1)\). For any constant \(c\in\mathbb{R}\) let \({\mathcal G}_c\) denote the class of all nonnegative functions \(g\) on \(Z_+\) satisfying \(g(1)\geq c\) and \(g(n+ 1)\geq g(n)+ c\), \(n\geq 1\). For any \(g\in{\mathcal G}_c\) put \(S^g_n= S_n- g(n)\) \((n\geq 0)\) and let \(M^g_n= \max_{0\leq i\leq n} S^g_i\) \((n\geq 0)\). Let \({\mathcal T}\) denote the family of all stopping times \(\sigma\) for \((S_n)\), i.e. \(\sigma\) is a \(Z_4\cup\{\infty\}\)-valued random variable such that, for all \(n\geq 0\), the event \(\{\sigma\leq n\}\) is independent of \(\{\xi_{n+1},\xi_{n+2},\dots\}\). Let \({\mathcal S}^*\) denote the family of all df's \(G\) such that \(\overline G(x)> 0\), \(x\in\mathbb{R}\), and \[ \int^x_0\overline G(x- y)\overline G(y)\,dy\sim 2\overline G(x)\int^\infty_0\overline G(x)\,dx\quad\text{as }x\to\infty. \] Finally let, for any \(\sigma\in{\mathcal T}\) and \(g\in{\mathcal G}_c\), \[ H^g_\sigma(x):= \sum_{n\geq 1} P(\sigma\geq n)\overline F(x+ g(n)),\quad x\in\mathbb{R}. \] The main result of the present paper is the following Theorem. Assume \((*)\). (i) Given any \(c> 0\), \(P(M^g_\sigma> x)\geq (1+ o(1)) H^g_\sigma(x)\) as \(x\to\infty\) holds uniformly over all \(\sigma\in{\mathcal T}\) and \(g\in{\mathcal G}_c\). (ii) Suppose additionally that \(F\in{\mathcal S}^*\). Then, given any \(c> 0\), \(P(M^g_\sigma> x)= (1+ o(1)) H^g_\sigma(x)\) as \(x\to\infty\) holds uniformly over all \(\sigma\in{\mathcal T}\) and \(g\in{\mathcal G}_c\). This extends results of \textit{S. Asmussen} [Ann. Appl. Probab. 8, No. 2, 354--374 (1998; Zbl 0942.60034)] and of \textit{S. Foss} and \textit{S. Zachary} [ibid. 13, No. 1, 37--53 (2003; Zbl 1045.60039)].
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