An asymptotic expansion for the distribution of the supremum of a random walk

Sgibnev, M.

Studia Mathematica, Tome 141 (2000), p. 41-55 / Harvested from The Polish Digital Mathematics Library

Résumé

Let ${S_n}$ be a random walk drifting to -∞. We obtain an asymptotic expansion for the distribution of the supremum of ${S_n}$ which takes into account the influence of the roots of the equation $1-∫_ℝe^{sx}F(dx)=0,F$ being the underlying distribution. An estimate, of considerable generality, is given for the remainder term by means of submultiplicative weight functions. A similar problem for the stationary distribution of an oscillating random walk is also considered. The proofs rely on two general theorems for Laplace transforms.

Détail

Publié le : 2000-01-01

Zbl 0962.60019

EUDML-ID : urn:eudml:doc:216755

Bibliographie

[00000] [1] G. Alsmeyer, On generalized renewal measures and certain first passage times, Ann. Probab. 20 (1991), 1229-1247. | Zbl 0759.60088

[00001] [2] J. Bertoin and R. A. Doney, Some asymptotic results for transient random walks, J. Appl. Probab. 28 (1996), 206-226. | Zbl 0854.60069

[00002] [3] A. A. Borovkov, Stochastic Processes in Queueing Theory, Springer, New York, 1976. | Zbl 0319.60057

[00003] [4] A. A. Borovkov, A limit distribution for an oscillating random walk, Theory Probab. Appl. 25 (1980), 649-651.

[00004] [5] A. A. Borovkov and D. A. Korshunov, Probabilities of large deviations for one-dimensional Markov chains. I. Stationary distributions, Theory Probab. Appl. 41 (1996), 1-24. | Zbl 0888.60025

[00005] [6] P. Embrechts and N. Veraverbeke, Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance Math. Econom. 1 (1982), 55-72. | Zbl 0518.62083

[00006] [7] W. Feller, An Introduction to Probability Theory and Its Applications II, Wiley, New York, 1966. | Zbl 0138.10207

[00007] [8] B. B. van der Genugten, Asymptotic expansions in renewal theory, Compositio Math. 21 (1969), 331-342. | Zbl 0211.21801

[00008] [9] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Colloq. Publ. 31, Amer. Math. Soc., Providence, 1957. | Zbl 0078.10004

[00009] [10] S. Janson, Moments for first passage times, the minimum, and related quantities for random walks with positive drift, Adv. Appl. Probab. 18 (1986), 865-879. | Zbl 0612.60060

[00010] [11] J. H. B. Kemperman, The oscillating random walk, Stochastic Process. Appl. 2 (1974), 1-29. | Zbl 0326.60081

[00011] [12] J. Kiefer and J. Wolfowitz, On the characteristics of the general queueing process, with applications to random walk, Ann. Math. Statist. 27 (1956), 147-161. | Zbl 0070.36602

[00012] [13] V. I. Lotov, Asymptotics of the distribution of the supremum of consecutive sums, Mat. Zametki 38 (1985), 668-678 (in Russian).

[00013] [14] B. A. Rogozin and M. S. Sgibnev, Banach algebras of measures on the line, Siberian Math. J. 21 (1980), 265-273. | Zbl 0457.46024

[00014] [15] M. S. Sgibnev, On the distribution of the supremum of sums of independent summands with negative drift, Mat. Zametki 22 (1977), 763-770 (in Russian).

[00015] [16] M. S. Sgibnev, Submultiplicative moments of the supremum of a random walk with negative drift, Statist. Probab. Lett. 32 (1997), 377-383. | Zbl 0903.60055

[00016] [17] M. S. Sgibnev, Equivalence of two conditions on singular components, ibid. 40 (1998), 127-131. | Zbl 0942.60023

[00017] [18] M. S. Sgibnev, On the distribution of the supremum in the presence of roots of the characteristic equation, Teor. Veroyatnost. i Primenen. 43 (1998), 383-390 (in Russian).

[00018] [19] R. L. Tweedie, The existence of moments for stationary Markov chains, J. Appl. Probab. 20 (1983), 191-196. | Zbl 0513.60067

[00019] [20] N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Process. Appl. 5 (1977), 27-37. | Zbl 0353.60073