Asymptotic Law of the 𝑗th Records in the Bivariate Exponential Case

Abstract

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 sequence of   r.v’s. uniformly distributed on ; then one has the following.

Theorem 5 (Deheuvels [12]). For all and for all   . for ,

Theorem 6 (Deheuvels [12]). For all , and for all   . for large enough,

Corollary 7. Let be a sequence of   r.v’s. with repartition function and . Then, a.s. for , , , Moreover, ., for , both members of these two inequalities are not satisfied infinitely many times.

Corollary 8. If , then , .

Theorem 9 (Barndorff-Nielsen [13]). If then   . with an increasing sequence of numbers.

Lemma 10. If is a sequence of pairs of r.v’s. with joint cumulative distribution function , then

Proof. The proof is obvious.

Theorem 11 (Marshall and Olkin [7]). If then , .

Theorem 12 (Galombos [6]). Let and be two sequences of r.v’s.; suppose that there exists a distribution function such that at any point of continuity of   Suppose that for all Then at all points of continuity of .

3. Main Result

Throughout the following, we consider as a sequence of   r.v’s. with joint cumulative distribution function whose marginals are, respectively, , and , .

3.1. Study of the Number of Coincidences

Let and be the sequences of th record times associated, respectively, to and , with and dependent, suggesting that every time , for any and , there is dependence of the th corresponding records.

Lemma 13. The number of coincidences, which means the number of times and belong, respectively, to the sets and , is finite.

Proof. Let us study the nature of the following series where is the event for which .
As is equivalent to , studying the nature of the series is equivalent to the study of the series with Let be a sequence of   r.v’s. with repartition and the sequence of   r.v’s. uniformly distributed on with and are identically distributed.
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 .
Choosing , , we obtain a lower bound of .
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 is a general term of Bertrand convergent series.
Hence we get which implies that Consequently, we have Using Corollary 8, we get .
Since implies it suffices to study the nature of the series to know that of , where is given by Using Theorem 11 of Marshall and Olkin, we get with (Riemann’s series).
Then 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 .

3.2. Some Record Series

Let us consider and give some properties of the following sequences:

Lemma 14. Consider the following

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 By the same method, we have

Now, let us consider the two following sequences: with

Lemma 16. Consider the following

Proof. Since the number of coincidences is finite, there exists finite, such that, for all ; is the sum of     r.v’s.
Between and we have coincidences and noncoincidences, so we must have noncoincidences between and ; hence This implies that with finite, so is a finite sum of r.v’s., which proves our result.
By the same method, we have

Lemma 17. and are individually the sum of    exponential r.v’s. with parameter so by the central limit theorem one gets

3.3. Study of the th Record Times

Definition 18. Let be a measurable space and be an increasing family of sub--algebra of , defined on an interval of or .
According to     .   defined on with values in is called a stopping time if

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 , that means an event definable by the behavior of the phenomenon before time .

Lemma 19. The sequence of th record times is a sequence of stopping time.

Proof. Considering the sub--algebra and the increasing family of sub--algebra of , is a stopping time if and only if But is equivalent to .
is measurable and included in because the ordered statistic is exhaustive.
But is included in which is equal to .
Consequently which implies that is a stopping time.
By the same method is also a stopping time.

3.4. Asymptotic Independence of and

In this section we will prove the asymptotic independence of and sequences of records issued from and , respectively.

Proposition 20. Let be a bivariate r.v’s. of records issued from pairs of r.v’s. with joint cumulative distribution whose marginals and are exponential with parameter ; then where is the standard normal distribution.

Proof. Since there exists an from which there is no coincidences, throughout the following, we will consider the case where we observe, first, a record for the .
being the algebra of events prior , all that has led to the appearance of that depends only on what has been observed before .
Let . Given and using Corollary 4, the first record is such that is independent of and fortiori of .
Thus, and are independent from all what has been observed before , including , which means it is independent of .
We conclude that(i)for all such that is asymptotically independent of with ,(ii)for all such that is asymptotically independent of with This sequential reasoning will be taken each time we have either a record for the or a record for the .
Thus, we prove that and are asymptotically independent and consequently and will be as well.
Indeed, if we denote, for the sake of simplicity, by ,we get Using (35), (39), and Theorem 12 Using (46), (49), and Theorem 12 Now using (50) we obtain which proves our result.

4. Conclusion

In this paper we proved the asymptotic independence of the th records and in the exponential case (), from sequences and , respectively. In this case we know that and are the type Gumbell, respectively. It would be interesting to see if we can obtain the same result when and are Frechet or Weibull type.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.