Consider a continuous-time renewal risk model, in which every main claim induces a delayed by-claim. Assume that the main claim sizes and the inter-arrival times form a sequence of identically distributed random pairs, with each pair obeying a dependence structure, and so do the by-claim sizes and the delay times. Supposing that the main claim sizes with by-claim sizes form a sequence of dependent random variables with dominatedly varying tails, asymptotic estimates for the ruin probability of the surplus process are investigated, by establishing a weakly asymptotic formula, as the initial surplus tends to infinity.
Let X = {X(t) ∈ R^d, t ∈ R^N} be a centered space-anisotropic Gaussian random field whose components satisfy some mild conditions. By introducing a new anisotropic metric in R^d, we obtain the Hausdorff and packing dimension in the new metric for the image of X. Moreover, the Hausdorff dimension in the new metric for the image of X has a uniform version.
Let X（1） = {X（1）（s）, s ∈ R＋} and X（2） = {X（2）（t）, t ∈ R＋} be two inde-pendent nondegenerate diffusion processes with values in Rd. The existence and fractal dimension of intersections of the sample paths of X （1） and X （2） are studied. More gener-ally, let E1, E2？（0,∞） and F ？Rd be Borel sets. A necessary condition and a suffcient condition for P{X（1）（E1）∩X（2）（E2）∩F 6=？}〉0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 × E2 × F in the metric space （R＋× R＋× Rd,ρb）, whereρb is an unsymmetric metric defined in R＋× R＋× Rd. Under reasonable conditions, results resembling those of Browian motion are obtained.
