A NOTE ON THE MAXIMUM OF A RANDOM-WALK

Abstract

For an integer-valued, aperiodic random walk (S-n)(n greater than or equal to 1) with negative drift we consider its maximum M and M(+), its maximum before first attaining a nonpositive value. As a converse to well-known results on the asymptotic behavior of m(n) = P(M greater than or equal to n) and r(n) = P(M(+) greater than or equal to n), it is shown that any of the relations m(n) similar to cr(-n) or r(n) similar to cr(-n) (c > 1, r > 1) implies that E(r(S1)) = 1.