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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
##### Keywords:
limit theorems; random walks; stable laws
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