## Local probabilities for random walks with negative drift conditioned to stay nonnegative.(English)Zbl 1307.60050

Summary: Let $$\{S_n: n\geq 0\}$$ with $$S_0=0$$ be a random walk with negative drift and let $$\tau_x=\min\{k>0: S_k<-x\}$$, $$x\geq 0$$. Assuming that the distribution of i.i.d. increments of the random walk is absolutely continuous with subexponential density we find the asymptotic behavior, as $$n\to \infty$$, of the probabilities $$\operatorname{P}(\tau_x=n)$$ and $$\operatorname{P}(S_n\in [y,y+\Delta), \tau_x>n)$$ for fixed $$x$$ and various ranges of $$y$$. The case of lattice distribution of increments is considered as well.

### MSC:

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