Gumbel And Frechet Convergence Of The Maxima Of Independent Random Walks

We consider point process convergence for sequences of independent and identically distributed random walks. The objective is to derive asymptotic theory for the largest extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Frechet distributions. The proofs depend heavily on precise large deviation results for sums of independent random variables with a finite moment generating function or with a subexponential distribution.