Abstract
Let {S n ; n ≥ 0} be an asymptotically stable random walk and let M n denote it’s maximum in the first n steps. We show that the asymptotic behaviour of local probabilities for M n can be approximated by the density of the maximum of the corresponding stable process if and only if the renewal mass-function based on ascending ladder heights is regularly varying at infinity. We also give some conditions on the random walk, which guarantee the desired regularity of the renewal mass-function. Finally, we give an example of a random walk, for which the local limit theorem for M n does not hold.
References
Aleshkyavichene A.K.: Local theorems for the maximum of sums of independent identically distributed random variables. Lit. Matem. Sb. 13, 163–174 (1973)
Alili L., Doney R.A.: Wiener–Hopf factorization revisited and some applications. Stoc. Stoc. Rep. 66, 87–102 (1999)
Bertoin J., Doney R.A.: On the local behaviour of ladder height distributions. J. Appl. Prob. 31, 816–821 (1994)
Caravenna F., Chaumont L.: Invariance principles for random walks conditioned to stay positive. Ann. Inst. H. Poincare Probab. Statist. 44, 170–190 (2008)
Doney R.A.: One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Relat. Fields 107, 451–465 (1997)
Doney R.A., Savov M.: The asymptotic behaviour of densities related to the supremum of a stable process. Ann. Probab. 38, 316–326 (2010)
Garsia A., Lamperti J.: A discrete renewal theorem with infinite mean. Comm. Math. Helv. 37, 221–234 (1963)
Jones, E.M.: Large deviations for random walks and Levy processes. PhD thesis, Manchester (2008)
Nagaev S.V.: An estimate of the rate of convergence of the distribution of the maximum of the sums of independent random variables. Sib. Math. J. 10, 443–458 (1969)
Nagaev S.V., Eppel M.S.: On a local limit theorem for the maximum of sums of independent random variables. Theory Probab. Appl. 21, 384–385 (1976)
Vatutin V.A., Wachtel V.: Local probabilities for random walks conditioned to stay positive. Probab. Theory Relat. Fields 143, 177–217 (2009)
Williamson J.A.: Random walks and Riesz kernels. Pacific J. Math. 25, 393–415 (1968)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wachtel, V. Local limit theorem for the maximum of asymptotically stable random walks. Probab. Theory Relat. Fields 152, 407–424 (2012). https://doi.org/10.1007/s00440-010-0326-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-010-0326-3
Keywords
- Limit theorems
- Random walks
- Renewal theorem
Mathematics Subject Classification (2000)
- 60G50
- 60G52