Asymptotic Law of the jth Records in the Bivariate Exponential Case

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(j) k } k⩾1 and {S(j) k } k⩾1 by the sequences of the jth 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(j) k } k⩾1 and {S(j) k } k⩾1 using the property of stopping time of the jth record times and that of the exponential distribution.


Introduction
Let (  ) 1⩽⩽ be a sequence of independent and identically distributed (..) random variables (r.v's.) from a distribution .Let us consider the th record time defined recurrently for  = 2, 3, . . .and  ⩾ 1 as ; where  , denotes the th order statistic of a sample of size .

Preliminaries
In this section we recall some relevant results for future use.
An ..bounding of  , is equivalent to an ..bounding of  , .
Using Theorems 5 and 6, and Corollary 7 of Deheuvels and Theorem 9 of Barndorff-Nielsen we obtain the a.s.bounding of  −+1, .
Choosing   = 1 − (1/(log ) (1+)/(−+1) ),  > 0, we obtain a lower bound of  −+1 .Indeed Let   be the general term of the series in the right side of the above equality.It is clear that with For  large enough   ∼   with   = 1/(log ) 1+ is a general term of Bertrand convergent series.Hence we get which implies that Consequently, we have Using Corollary 8, we get implies it suffices to study the nature of the series ∑  P(   ) to know that of ∑  P(   ), where    is given by Using Theorem 11 of Marshall and Olkin, we get By Borel-Cantelli's Lemma, which means that with probability one at most a finite number of   are realized simultaneously and consequently that the number of coincidences is finite ..

Some Record Series.
Let us consider and give some properties of the following sequences: Proof.Consider the following with Hence where  is the set of  for which there are coincidences; as this set is finite ∑ ∈  ()  < ∞.By the same method we have Lemma 15.Consider Proof.Consider the following Using Corollary 3 and Theorem 12 of Galombos and the central limit theorem we get lim (50)

Study of the 𝑗th Record Times
Definition 18.Let (Ω, F) be a measurable space and (F  ) ∈ be an increasing family of sub--algebra of F, defined on an interval  of R or N.
According to (F  ) ∈  .V.  defined on (Ω, F) with values in  is called a stopping time if { :  () ⩽ } ∈ F  , ∀ ∈ . (51) Interpretation.The notion of stopping time corresponds to the notion of stopping some process at a random time .Indeed, the stopping time is a function of .Also the definition (51) expresses that all the contingencies that lead to the previous stopping time are an event of F  , that means an event definable by the behavior of the phenomenon before time .