A local limit theorem for random walk maxima with heavy tails
For a random walk with negative mean and heavy-tailed increment distribution F, it is well known that under suitable subexponential assumptions, the distribution pi of the maximum has a tail pi(x, infinity) which is asymptotically proportional to integral(x)(infinity)F(y,infinity) dy. We supplement here this by a local result showing that pi(x, x + z] is asymptotically proportional to zF(x,infinit