Geodesic
On the distribution of the supremumof a random walk when the characteristic equation has roots
Teoriâ veroâtnostej i ee primeneniâ, Tome 43 (1998) no. 2, pp. 383-390 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the random walk $\{S_{n}\}$, generated by a sequence $\{X_{k}\}$ of independent identically distributed random variables with ${\mathbf{E}}X_{1}\in (-\infty,0)$. The influence of the roots of the characteristic equation $1-{\mathbf{E}}\exp(sX_{1})=0$ in the analyticity strip of the Laplace transform ${\mathbf{E}}\exp(sX_{1})$ on the distribution of the supremum $\sup_{n\ge 0}S_{n}$ is studied. An analogous problem is investigated for the stationary distribution of an oscillating random walk.
Keywords: random walk, supremum, roots of the characteristic equation, absolutely continuous component, oscillating random walk, stationary distribution, asymptotic behavior. 
@article{TVP_1998_43_2_a13,
     author = {M. S. Sgibnev},
     title = {On the distribution of the supremumof a~random walk when the characteristic equation has roots},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {383--390},
     year = {1998},
     volume = {43},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1998_43_2_a13/}
}
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M. S. Sgibnev. On the distribution of the supremumof a random walk when the characteristic equation has roots. Teoriâ veroâtnostej i ee primeneniâ, Tome 43 (1998) no. 2, pp. 383-390. http://geodesic.mathdoc.fr/item/TVP_1998_43_2_a13/