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Short Communications

On the distribution of the supremumof a random walk when the characteristic equation has roots

M. S. Sgibnev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: 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.

Received: 05.12.1997

DOI: 10.4213/tvp1475

 English version: Theory of Probability and its Applications, 1999, 43:2, 322–329

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