Asymptotic probabilities of an exceedance over renewal thresholds with an application to risk theory

Abstract

Let (Y_n, N_n)_n≥1 be independent and identically distributed bivariate random variables such that the N_n are positive with finite mean ν and the Y_n have a common heavy-tailed distribution F. We consider the process (Z_n)_n≥1 defined by Z_n = Y_n - Σ_n-1, where Σ_n-1 = ∑_k=1^n-1N_k. It is shown that the probability that the maximum M = max_n≥1Z_n exceeds x is approximately ν^-1∫_x^∞F'(u)du, as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.