## Exponential moments of first passage times and related quantities for random walks.(English)Zbl 1235.60118

Let $$(X_{n},n\geq 1)$$ denote a sequence of i.i.d. real-valued random variables, and let $$S_{0}=0$$, $$S_{n+1}=S_{n}+X_{n+1}$$, $$n\geq 0$$. For real $$x$$, let $$\tau (x)$$ denote the first passage time into $$(x,\infty )$$, let $$N(x)$$ denote the number of visits to the interval $$(-\infty ,x)$$ and let $$\rho (x)$$ denote the last exit time from $$(-\infty ,x)$$. If $$X_{1}>0$$, we have $$\tau (x)-1=N(x)=\rho (x)$$, $$x\geq 0$$. For general real valued random variables with $$\text{P}(X<0)>0$$, the authors provide if-and-only-if conditions to ensure that $$\text{E}(\exp (a\tau (x))<\infty$$ for $$a>0$$ and prove similar results for $$N(x)$$ and $$\rho (x)$$. In the second main result, the authors obtain conditions under which $$\text{E}(\exp (a\tau (x))\thicksim c\exp (\gamma x)$$, where $$\gamma$$ and $$c$$ are explicitly given. They obtain similar results for $$N(x)$$ and $$\rho (x)$$. The paper closes with some well chosen examples.

### MSC:

 60K05 Renewal theory 60G40 Stopping times; optimal stopping problems; gambling theory 60F99 Limit theorems in probability theory
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