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Local probabilities for random walks conditioned to stay positive
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  • Published: 04 January 2008

Local probabilities for random walks conditioned to stay positive

  • Vladimir A. Vatutin1 &
  • Vitali Wachtel2 

Probability Theory and Related Fields volume 143, pages 177–217 (2009)Cite this article

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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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Author information

Authors and Affiliations

  1. Steklov Mathematical Institute RAS, Gubkin street 8, 19991, Moscow, Russia

    Vladimir A. Vatutin

  2. Technische Universität München, Zentrum Mathematik, Bereich M5, 85747, Garching bei München, Germany

    Vitali Wachtel

Authors
  1. Vladimir A. Vatutin
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  2. Vitali Wachtel
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Corresponding author

Correspondence to Vladimir A. Vatutin.

Additional information

Supported by the Russian Foundation for Basic Research grant 08-01-00078 and by the GIF.

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Cite this article

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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  • Received: 16 February 2007

  • Revised: 25 November 2007

  • Published: 04 January 2008

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0124-8

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Keywords

  • Limit theorems
  • Random walks
  • Stable laws

Mathematics Subject Classification (2000)

  • 60G50
  • 60G52
  • 60E07
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