An invariance principle in k-dimensional extended renewal theory

Abstract

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ ) v, τ= 1,· ··, k is the covariance matrix of In this paper the weak convergence of Z n in (D[0, ∞)) k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t 1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ 1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.