Asymptotic Law of the jth Records in the Bivariate Exponential Case

Abstract

We consider a sequence (Xi,Yi)1⩽i⩽n of independent and identically distributed random variables with joint cumulative distribution H(x,y), which has exponential marginals F(x) and G(y) with parameter λ=1. We also assume that Xi(ω)≠Yi(ω), ∀i∈N, and ω∈Ω. We denote Rk(j)k⩾1 and Sk(j)k⩾1 by the sequences of the jth records in the sequences (Xi)1⩽i⩽n, (Yi)1⩽i⩽n, respectively. The main result of of the paper is to prove the asymptotic independence of Rk(j)k⩾1 and Sk(j)k⩾1 using the property of stopping time of the jth record times and that of the exponential distribution.