ON THE DISTRIBUTION OF THE SUPREMUM OF A RANDOM WALK WHEN THE CHARACTERISTIC EQUATION HAS ROOTS.

We consider the random walk {S[SUBn]}, generated by a sequence {X[SUBk]} of independent identically distributed random variables with EX[SUB1] &epsis; (-∞, 0). The influence of the roots of the characteristic equation 1-E exp([SUBs]X[SUB1]) = 0 in the analyticity strip of the Laplace transform E exp([SUBs]X[SUB1]) on the distribution of the supremum sup[SUBn≥0] S[SUBn] is studied. An analogous problem is investigated for the stationary distribution of an oscillating random walk. [ABSTRACT FROM AUTHOR]