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

Studia Mathematica (2000)

• Volume: 140, Issue: 1, page 41-55
• ISSN: 0039-3223

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## Abstract

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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 $1-{\int }_{ℝ}{e}^{sx}F\left(dx\right)=0,F$ 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.

## How to cite

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Sgibnev, M.. "An asymptotic expansion for the distribution of the supremum of a random walk." Studia Mathematica 140.1 (2000): 41-55. <http://eudml.org/doc/216755>.

@article{Sgibnev2000,
abstract = {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.},
author = {Sgibnev, M.},
journal = {Studia Mathematica},
keywords = {random walk; supremum; submultiplicative function; characteristic equation; absolutely continuous component; oscillating random walk; stationary distribution; asymptotic expansions; Banach algebras; Laplace transform},
language = {eng},
number = {1},
pages = {41-55},
title = {An asymptotic expansion for the distribution of the supremum of a random walk},
url = {http://eudml.org/doc/216755},
volume = {140},
year = {2000},
}

TY - JOUR
AU - Sgibnev, M.
TI - An asymptotic expansion for the distribution of the supremum of a random walk
JO - Studia Mathematica
PY - 2000
VL - 140
IS - 1
SP - 41
EP - 55
AB - 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.
LA - eng
KW - random walk; supremum; submultiplicative function; characteristic equation; absolutely continuous component; oscillating random walk; stationary distribution; asymptotic expansions; Banach algebras; Laplace transform
UR - http://eudml.org/doc/216755
ER -

## References

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15. [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. [16] M. S. Sgibnev, Submultiplicative moments of the supremum of a random walk with negative drift, Statist. Probab. Lett. 32 (1997), 377-383. Zbl0903.60055
17. [17] M. S. Sgibnev, Equivalence of two conditions on singular components, ibid. 40 (1998), 127-131. Zbl0942.60023
18. [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).
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20. [20] N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Process. Appl. 5 (1977), 27-37. Zbl0353.60073

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