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Local probabilities for random walks with negative drift conditioned to stay nonnegative

   
@article{EJP3426,
	author = {Denis Denisov and Vladimir Vatutin and Vitali Wachtel},
	title = {Local probabilities for random walks with negative drift conditioned to stay nonnegative},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {19},
	year = {2014},
	keywords = {Random walk, negative drift, conditional local limit theorems, exit time},
	abstract = {Let $S_n, n=0,1,...,$ 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 $\mathbf{P}(\tau_x=n)$ and $\mathbf{P}(S_n\in (y,y+\Delta],\tau_x>n)$ for fixed $x$ and various ranges of $y.$ The case of lattice distribution of increments is considered as well.},
	pages = {no. 87, 1-17},
	issn = {1083-6489},
	doi = {10.1214/EJP.v19-3426},    
        url = {http://ejp.ejpecp.org/article/view/3426}}