SECOND ORDER APPROXIMATION FOR THE DISTRIBUTION OF THE MAXIMUM OF A RANDOM WALK WITH NEGATIVE DRIFT AND INFINITE VARIANCE

Let xi,xi(1),xi(2),... be independent identically distributed random variables, and set a := -E xi > 0, S-n := Sigma(n)(j= 1) xi(j), S-0 = 0, S := sup(n) >= 0 S-n, F+(t) := P(xi >= t), F-(t) := P(xi <= -t), F-+(I) (t) = integral(infinity)(t) F+(u) du. It is well known (see, e. g., [N. Veraverbeke, Stoch. Process. Appl., 5 (1977), pp. 27-37], [A. A. Borovkov, Probability Theory, Springer, New York, 2013]) that if the function FI+ (t) is subexponential, then P(S >= x) similar to F-+(I) (x)/ a as x -> infinity. Under the condition that E xi(2) < infinity and the function F+(x) is either regularly varying at infinity or semiexponential, the next term of the asymptotic expansion for P(S >= x) as x -> infinity (which is of the order of F+(x)) was found in [A. A. Borovkov, Sib. Math. J., 43 (2002), pp. 995-1022] (see also [A. A. Borovkov and K. A. Borovkov, Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions, Cambridge University Press, Cambridge, UK, 2008]). In the present paper we obtain second order approximation for P(S >= x) in the case where E xi(2) = infinity and the functions F-+/-(t) satisfy certain regular variation conditions. The results are extended to compound renewal processes.