Gumbel and Fréchet 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 Fréchet 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.

Precise Large Deviation of Claim Surplus Process in a Risk Model with END Claim Sizes

In this paper, we study a risk model in which the claim sizes are extended negatively dependent random variables with consistently varying tails, and the arrival of the successive insurance policies forms a nonstandard renewal processes. For this risk model, we give the precise large deviation of the claim surplus process.