Asymptotic Law of the th Records in the Bivariate Exponential Case

We consider a sequence of independent and identically distributed random variables with joint cumulative distribution , which has exponential marginals and with parameter . We also assume that , , and . We denote and by the sequences of the th records in the sequences , , respectively. The main result of of the paper is to prove the asymptotic independence of and using the property of stopping time of the th record times and that of the exponential distribution. 1. Introduction Let 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 and as and the th ordinary th record from as where denotes the th order statistic of a sample of size . In 1976 Dziubdziela-Kopocinski [1] proved that where is one of the three well known limit laws of (Frechet, Wiebull, and Gumbell), is the standard normal distribution, and , are the constants of normalization. Taking account of these three limit laws we get those of the th record; namely,(i)Type 1: (ii)Type 2: (iii)Type 3: The authors have presented the expressions of the probability density function and the distribution function of the th record as follows: Contrary to that of records, the theory of limit law of the maximum has been extended to the bivariate case by the works of Finkelshtein [2], Geffroy [3], de Oliveira [4], Sibuya [5], Galombos [6], Marshall and Olkin [7], Kotz and Nadarajah [8], Coles [9], and Smith [10]. In this work, we assume that , and . Moreover, we will focus on the bivariate r.v’s.？？ issued from pairs of r.v’s.？？ with joint cumulative distribution whose marginals and are exponential with parameter and prove that where is the standard normal distribution, and , . 2. Preliminaries In this section we recall some relevant results for future use. Theorem 1 (Deheuvels [11]). Let be a sequence of ？ r.v’s. with continues repartition function. Let and be, respectively, the sequences of order statistics and th records. and for all one has the following.(i)For and , the sequences ？form a Markov chain with equal probability of transition. That means (ii)The assertions and are equivalent. Corollary 2. Let be a sequence of ？ r.v’s. with repartition continues and let, for fixed, be the sequence of th records from . If is a fixed integer, and are independent if and only if with for and being positive finite constants. Corollary 3. For and being the sequence of th records, then is a sequence of ？ r.v’s.s with repartition function . Corollary 4. For , and for all , is independent of . Let be the