Abstract:
Let $S_{0}=0,\{S_{n},\,n\geq 1\}$ be a random walk generated by a sequence
of i.i.d. random variables $X_{1},X_{2},...$ 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 \lbrack x,x+\Delta )|\tau ^{-}>n)$ with fixed $\Delta $
and $x=x(n)\in (0,\infty )$ $.$
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