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Local probabilities for random walks conditioned to stay positive. (English) Zbl 1158.60014
Authors’ summary: Let \(S_0=0\), \(\{S_n\), \(n\geq 1\}\) be a random walk generated by a sequence of i.i.d. random variables \(X _1, X _2,\dots\) and let \(\tau^-=\min \{n\geq 1:S_n\leq 0\}\) and \(\tau^+=\min\{n\geq 1:S_n> 0\}\). Assuming that the distribution of \(X_1\) belongs to the domain of attraction of an \(\alpha \)-stable law we study the asymptotic behavior, as \(n\rightarrow \infty\), of the local probabilities \({\mathbf P}(\tau^\pm=n)\) and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities \({\mathbf P}(S_{n}\in [x,x+\Delta )\mid \tau^-> n)\) with fixed \(\Delta\) and \(x=x(n)\in (0,\infty)\).

MSC:
60G50 Sums of independent random variables; random walks
60G52 Stable stochastic processes
60E07 Infinitely divisible distributions; stable distributions
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