Asymptotic Law of the j th Records in the Bivariate Exponential Case

الملخص EN

We consider a sequence ( X i , Y i ) 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 X i ( ω ) ≠ Y i ( ω ) , ∀ i ∈ N , and ω ∈ Ω .

We denote R k ( j ) k ⩾ 1 and S k ( j ) k ⩾ 1 by the sequences of the j th records in the sequences ( X i ) 1 ⩽ i ⩽ n , ( Y i ) 1 ⩽ i ⩽ n , respectively.

The main result of of the paper is to prove the asymptotic independence of R k ( j ) k ⩾ 1 and S k ( j ) k ⩾ 1 using the property of stopping time of the j th record times and that of the exponential distribution.