# 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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topSgibnev, 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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