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

Denisov, Denis; Vatutin, Vladimir; Wachtel, Vitali

Electron. J. Probab. 19, Paper No. 88, 17 p. (2014).

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

Keywords:

random walk; negative drift; conditional local limit theorems; exit time