Abstract
Let S 0 = 0, {S n , n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X 1, X 2, . . . and let \(\tau ^{-}={\rm min} \{ n \geq 1:S_{n}\leq 0 \}\) and \(\tau ^{+}={\rm min}\{n\geq1:S_{n} > 0\} \). Assuming that the distribution of X 1 belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as \({n\rightarrow \infty }\), of the local probabilities \({\bf P}{(\tau ^{\pm }=n)}\) and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities \({\bf P}{(S_{n} \in [x,x+\Delta )|\tau^{-} > n)}\) with fixed Δ and \({x=x(n)\in (0,\infty )}\).
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Supported by the Russian Foundation for Basic Research grant 08-01-00078 and by the GIF.
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Vatutin, V.A., Wachtel, V. Local probabilities for random walks conditioned to stay positive. Probab. Theory Relat. Fields 143, 177–217 (2009). https://doi.org/10.1007/s00440-007-0124-8
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DOI: https://doi.org/10.1007/s00440-007-0124-8