Electronic Communications in Probability

Large deviation results for random walks conditioned to stay positive

Ronald Doney and Elinor Jones

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Abstract

Let $X_{1},X_{2},...$ denote independent, identically distributed random variables with common distribution $F$, and $S$ the corresponding random walk with $\rho :=\lim_{n\rightarrow \infty }P(S_{n}>0)$ and $\tau :=\inf \{n\geq 1:S_{n}\leq 0\}$. We assume that $X$ is in the domain of attraction of an $\alpha $-stable law, and that $P(X\in \lbrack x,x+\Delta ))$ is regularly varying at infinity, for fixed $\Delta >0$. Under these conditions, we find an estimate for $P(S_{n}\in \lbrack x,x+\Delta )|\tau >n)$, which holds uniformly as $x/c_{n}\rightarrow \infty $, for a specified norming sequence $c_{n}$. This result is of particular interest as it is related to the bivariate ladder height process $((T_{n},H_{n}),n\geq 0)$, where $T_{r}$ is the $r$th strict increasing ladder time, and $H_{r}=S_{T_{r}}$ the corresponding ladder height. The bivariate renewal mass function $g(n,dx)=\sum_{r=0}^{\infty }P(T_{r}=n,H_{r}\in dx)$ can then be written as $g(n,dx)=P(S_{n}\in dx|\tau >n)P(\tau >n)$, and since the behaviour of $P(\tau >n)$ is known for asymptotically stable random walks, our results can be rephrased as large deviation estimates of $g(n,[x,x+\Delta))$.

Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 38, 11 pp.

Dates
Accepted: 28 August 2012
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465263171

Digital Object Identifier
doi:10.1214/ECP.v17-2282

Mathematical Reviews number (MathSciNet)
MR2970702

Zentralblatt MATH identifier
1252.60041

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G52: Stable processes 60E07: Infinitely divisible distributions; stable distributions

Keywords
Limit theorems Random walks Stable laws

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Doney, Ronald; Jones, Elinor. Large deviation results for random walks conditioned to stay positive. Electron. Commun. Probab. 17 (2012), paper no. 38, 11 pp. doi:10.1214/ECP.v17-2282. https://projecteuclid.org/euclid.ecp/1465263171


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References

  • Alili, L.; Doney, R. A. Wiener-Hopf factorization revisited and some applications. Stochastics Stochastics Rep. 66 (1999), no. 1-2, 87–102.
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2
  • Denisov, D.; Dieker, A. B.; Shneer, V. Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36 (2008), no. 5, 1946–1991.
  • Doney, R. A. A large deviation local limit theorem. Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 3, 575–577.
  • Doney, R. A. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 (1997), no. 4, 451–465.
  • Doney, R. A. Local behaviour of first passage probabilities. Probab. Theory Related Fields 152 (2012), no. 3-4, 559–588.
  • Doney, R. A.; Savov, M. S. The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab. 38 (2010), no. 1, 316–326.
  • Fuk, D. H.; Nagaev, S. V. Probabilistic inequalities for sums of independent random variables. (Russian) Teor. Verojatnost. i Primenen. 16 (1971), 660–675.
  • Jones, E. M.: PhD thesis, The University of Manchester, (2008).
  • Nagaev, S. V. On the asymptotic behavior of probabilities of one-sided large deviations. (Russian) Teor. Veroyatnost. i Primenen. 26 (1981), no. 2, 369–372.
  • Pruitt, William E. The growth of random walks and Lévy processes. Ann. Probab. 9 (1981), no. 6, 948–956.
  • Tchachkuk, S. G.: Limit theorems for sums of independent random variables belonging to the domain of attraction of a stable law. Candidate's dissertation, Tashent (in Russian), (1977).
  • Vatutin, Vladimir A.; Wachtel, Vitali. Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 (2009), no. 1-2, 177–217.
  • Wachtel, Vitali. Local limit theorem for the maximum of asymptotically stable random walks. Probab. Theory Related Fields 152 (2012), no. 3-4, 407–424.