A Note on the Severity of Ruin in the Renewal Model with Claims of Dominated Variation

This paper investigates the tail asymptotic behavior of the severity of ruin (the deficit at ruin) in the renewal model. Under the assumption that the tail probability of the claimsize is dominatedly varying, a uniform asymptotic formula for the tail probability of the deficit at ruin is obtained. 1. Model and main result Throughout this paper, for any 0 ≤ a < b < ∞ the integral symbol ∫ b a is understood as ∫ (a,b] but ∫ ∞ a = ∫ (a,∞) . For a given distribution function (d.f.) F with finite mean μ and support [0,∞), we denote its tail by F (x) = 1 − F (x) and its equilibrium d.f. by (1.1) Fe(x) = 1 μ ∫ x 0 F (t)dt, x ≥ 0. The symbol F ∗n represents the n-fold convolution of F with F ∗0 being degenerate at 0. The following renewal risk model has received extensive attention in risk theory; see Embrechts et al. [2], Rolski et al. [9] and Asmussen [1] for reviews. In this model the successive claims, Zk, k ≥ 1, form a sequence of independent, identically distributed (i.i.d.) and nonnegative random variables (r.v.’s) with common d.f. F and finite mean μ. Their occurrence times, σk, k ≥ 1, comprise a renewal process N(t) = #{k ≥ 1;σk ∈ (0, t]}, t ≥ 0, i.e. that the inter-occurrence times θ1 = σ1, θk = σk − σk−1, k ≥ 2, are i.i.d. non-negative r.v.’s. Suppose Received June 20, 2002. 2000 Mathematics Subject Classification: 62P05, 60K05.