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ArticleOriginal scientific text

Title

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

Authors 1

Affiliations

  1. Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 90, 630090 Russia

Abstract

Let {Sn} be a random walk drifting to -∞. We obtain an asymptotic expansion for the distribution of the supremum of {Sn} which takes into account the influence of the roots of the equation 1-∫ℝesxF(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.

Keywords

ENG
random walk, supremum, submultiplicative function, characteristic equation, absolutely continuous component, oscillating random walk, stationary distribution, asymptotic expansions, Banach algebras, Laplace transform

Bibliography

  1. G. Alsmeyer, On generalized renewal measures and certain first passage times, Ann. Probab. 20 (1991), 1229-1247.
  2. J. Bertoin and R. A. Doney, Some asymptotic results for transient random walks, J. Appl. Probab. 28 (1996), 206-226.
  3. A. A. Borovkov, Stochastic Processes in Queueing Theory, Springer, New York, 1976.
  4. A. A. Borovkov, A limit distribution for an oscillating random walk, Theory Probab. Appl. 25 (1980), 649-651.
  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.
  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.
  7. W. Feller, An Introduction to Probability Theory and Its Applications II, Wiley, New York, 1966.
  8. B. B. van der Genugten, Asymptotic expansions in renewal theory, Compositio Math. 21 (1969), 331-342.
  9. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Colloq. Publ. 31, Amer. Math. Soc., Providence, 1957.
  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.
  11. J. H. B. Kemperman, The oscillating random walk, Stochastic Process. Appl. 2 (1974), 1-29.
  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.
  13. V. I. Lotov, Asymptotics of the distribution of the supremum of consecutive sums, Mat. Zametki 38 (1985), 668-678 (in Russian).
  14. B. A. Rogozin and M. S. Sgibnev, Banach algebras of measures on the line, Siberian Math. J. 21 (1980), 265-273.
  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).
  16. M. S. Sgibnev, Submultiplicative moments of the supremum of a random walk with negative drift, Statist. Probab. Lett. 32 (1997), 377-383.
  17. M. S. Sgibnev, Equivalence of two conditions on singular components, ibid. 40 (1998), 127-131.
  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).
  19. R. L. Tweedie, The existence of moments for stationary Markov chains, J. Appl. Probab. 20 (1983), 191-196.
  20. N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Process. Appl. 5 (1977), 27-37.
Studia Mathematica
2000/140/1
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Pages:
41-55
Main language of publication
English
Received
1999-03-16
Accepted
2000-02-07
Published
2000
Exact and natural sciences