# Limit Distributions of Ordered Random Variables in Mixture of Two Gaussian Sequences

Authors: H. M. Barakat; M. H. Dwes
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study the limit distributions of extreme, intermediate, and central order statistics, as well as record values, of the mixture of two stationary Gaussian sequences under equicorrelated setup.


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## Body

## 1. Introduction

Because the mixture model, which is a convex combination of two distribution functions (DFs) and incorporates the qualities of the separate DFs, it is a strong and flexible tool for modelling complex data (e.g., see Barakat [1] and Barakat et al. [2]). Apart from this benefit, mixture models are widely employed in a variety of applications, as evidenced by Titterington et al. [3] and Doğru and Arslan [4]. For example, a two-dimensional mixture model arises in the theory of reliability when individuals belong to one of two distinct types, with proportions p and q=1−p, of two populations which have lifetime DFs FX1 and FX2. An individual is randomly selected from the population and then has the lifetime DF:(1)FXx=pFX1x+qFX2x.It is known that model (1) is not derivable (cf. Nelson [5]), i.e., there does not exist any two-place function X.,. such that, for any pair X1,X2 of random variables (RVs) with respective DFs FX1 and FX2, the mixture pFX1+qFX2 is the DF of the RV XX1,X2. However, the mixture model (1) can be represented in terms of RVs rather than DFs (cf. Bernard and Vanduffel [6] and Barakat et al. [2]) as(2)X=IpX1+IqX2,where Ip is a Bernoulli distributed RV with parameter p, q=1−p, Iq=1−Ip, and the RVs X1 and X2 are independent of Ip (and Iq). Representation (2) was recently used in literature to derive the quantile function of the mixture model (1) (e.g., Barakat et al. [2]). On the other hand, this representation will be an essential pillar of our study. Our objective is to study the limit distributions of important ordered RVs, which are order statistics (OSs) (extreme, intermediate, and central OSs) and record values of a mixture of two stationary Gaussian sequences (SGSs) under an equicorrelated setup.Despite the fact that we consider the equicorrelated setup, the paper’s results are of considerable theoretical and practical interest due to the remarkable paucity of works on the limit of OSs and record values arising from the mixture model and the absence of any work concerning the mixture model of two SGSs. Moreover, there are some scientific evidences that a deviation from this “neat”constant sequence of the correlation will not dramatically affect the limiting form of the statistics that are considered in this paper. Actually, for the maximum OSs, based on SGS, a deviation from constant to varying correlation does not affect the limiting form of the maximum OSs when properly normalized (e.g., see Theorems 3.8.1 and 3.8.2 in Galambos [7]). For the limit DFs of order RVs under the equicorrelated setup, the literature is abounded with some variant works [8–14].LetX1,ii=1n and X2,ii=1n be two SGSs. Furthermore, for j=1,2, let Xj,ii=1n have zero mean, unit variance, and constant correlation coefficient rj,n=EXj,iXj,k≥0,i≠k, written Xj,i∼Gas0,1,rj,n. The sequence Xj,ii=1n,j=1,2, can be exemplified by Xj,i=rj,nYj,0+1−rj,nYj,i,i=1,2,…,n, where Yj,0,Yj,1,…,Yj,n are i.i.d standard normal variables (cf. [7]).It is worth mentioning that both representations (1) and (2) show that the dependence structure between the two RVs X1 and X2 does not influence these representations. Therefore, without any loss of generality, we assume that the two SGSs X1,ii=1n and X2,ii=1n are independent. Therefore, relying on representation (2), the mixture of these sequences is(3)Xi=IpX1,i+IqX2,i,i=1,2,…,n,where X1,ii=1n and X2,ii=1n are independent and each of them is independent of Ip (and Iq). On the other hand, sequence (3) can also be exemplified by(4)Xi=Z0+Zi,i=1,2,…,n,where(5)Z0=Ipr1,nY1,0+Iqr2,nY2,0,Zi=Ip1−r1,nY1,i+Iq1−r2,nY2,i,PZ0≤z=pΦz/r1,n+qΦz/r2,n, Φ. is the standard normal DF, and Z1,Z2,…Zn are i.i.d RVs with common DF as follows:(6)ℱz≔PZi≤z=pΦz1−r1,n+qΦz1−r2,n.Now, for any1≤s≤n, the s th OS based on the sequence Xii=1n, represented by (3), can be expressed as(7)Xs:n=Z0+Zs:n,where Zs:n is the s th OS based on the sequence Zii=1n, represented by (5).For fixed ranks≥1, with respect to n, Xs:n and Xsn′:n≔Xn−s+1:n are called the s th lower and upper extremes, respectively. Any result for the lower OSs may be inferred from the upper OSs using the well-known relation maxx1,x2,…,xn=−min−x1,−x2,…,−xn and vice versa. Besides extreme OSs, there are two types of OSs according to their rank nature. A sequence Xsn:n is called a sequence of OSs with variable rank if maxsn,n−sn+1⟶∞, as n⟶∞. Consequently, two particular variable ranks are of special interest:(1)
sn/n⟶0 (or sn/n⟶1), as n⟶∞, which will be referred to as the lower intermediate rank case (or the upper intermediate rank case).(2)
sn/n⟶λ0<λ<1, as n⟶∞, which we shall call the case of central ranks. A familiar example of the central OSs is the λ th sample quantile, where sn=λn,0<λ<1, and x denotes the largest integer not exceeding x.Record values in a sequence of RVs are successive maxima or values that strictly exceed all preceding values. LetXn,n≥1 be i.i.d RVs with common DF FXx. Then, Xj is an (upper) record value if Xj>Xi,∀i<j, and therefore X1 is a record value. The record time sequence Tn,n≥1 is times at which records appear. Therefore, T1=1 and Tn=minj:Xj>XTn−1,n>1. Thus, the record value sequence Rn is defined by Rn=XTn. For more details about the record value model, see Arnold et al. [15]. Thus, the records RnX based on the sequence Xii=1n, represented by (3), can be expressed as(8)RnX=Z0+RnZ,where RnZ is the record value based on the sequence Zii=1n, represented by (5).In the second section of this paper, we study the asymptotic distribution of upper extreme OSs concerning the mixture of two SGSs represented by (3), under mild conditions. In the third section, we obtain the parallel results for the central OSs. In the fourth section, the asymptotic behavior of the intermediate OSs of the mixture of two SGSs is studied. In the last section, the asymptotic behavior of the record in the mixture of two SGSs, represented in (8), is studied.Everywhere in what follows, the symbols⟶n, →nw, and ⟶np stand for convergence, converge weakly, and converge in probability, as n⟶∞, respectively. Also, ∗ stands for the convolution operation.


## 2. Limit Distributions of Extreme OSs in a Mixture of Two SGSs

We start with a slight generalization of a result of [16], which will be needed for our study in this section.Lemma 1.
LetFi. be a given sequence of nondegenerate DFs and suppose η1,η2,…,ηn are i.i.d RVs with mixture DF,(9)Fηx=∑i=1kpiFiαi,nx,∑i=1kpi=1,αi,n>0.Furthermore, letηsn′:n≔ηn−s+1:n be the sn′ th OS of the sequence η1,η2,…,ηn. Then, for some suitable linear transformation Gnx=anx+bn,an>0, we get(10)Fηsn′:nGnx≔Pηsn′:n≤Gnx∏i=1kΨipix∑l=0s−1−∑i=1kpilogΨixll!,if Fiαi,nGnx belongs to the max-domain of attraction of the nondegenerate max-type Ψix, written Fiαi,nGnx∈DΨix,∀i=1,2,…,k.Remark 1.
According to the Extremal Type theorem (cf. [7, 17]), there are only three possible max-types, namely, max-Weibull, Fréchet, and Gumbel types. The Gumbel type is an exclusively important type in our study in this section, Ψ3x=exp−exp−x, because of the well-known fact Φanx+bn∈DΨ3x, where an=2logn−1/2 and bn=1/an−an/2loglogn+log4π.Proof of Lemma1.
The proof follows by using Theorems 2.2.1 and 2.2.2 in [17] and the obvious relation:(11)n1−FηGnx=∑i=1kpin1−Fiαi,nGnx⟶nw−∑i=1kpilogΨix=−log∏i=1kΨipix.In [16], Lemma 1 was proved for s=1 and αi,n=1. Moreover, AL-Hussaini and El-Adll [16] claimed that the converse of Lemma 1 is true, but Sreehari and Ravi [18] proved that this allegation is false. Moreover, Barakat et al. [19] refined the results of [16, 18] by revealing that any scale normalizing constant an in Gx determines uniquely (up to scale changes) the max-stable type. The following theorem gives the asymptotic distribution of upper extreme OSs Xsn′:n≔Xn−s+1:n concerning sequence (3) (and consequently sequence (7)).Theorem 1.
Concerning sequence (7), let an and bn be defined as in Remark 1. Furthermore, let rj,nlogn⟶nτj≥0, j=1,2. If τ1=τ2=0, then(12)FXsn′:nanx+bn⟶nwexp−e−x∑l=0s−1e−xll!=e−ux∑l=0s−1ulxl!≔Hux,where ux=e−x whereas if maxτ1,τ2>0, then(13)FXsn′:nanx+bn⟶nwHpux+τ1+qux+τ2∗pΩx,τ1+qΩx,τ2,where(14)Ωx,τ≔Φx2τ,ifτ>0,I0,∞x,ifτ=0,where IAx is the indicator function of set A. On the other hand, let maxr1,nlogn,r2,nlogn⟶n∞. Then,(1)
FXsn′:nr2,nx+bn⟶nwpI0,∞x+qΦx if rj,nlogn⟶n∞, j=1,2, and r1,n/r2,n⟶n0, or if r1,nlogn⟶nτ1≥0 and r2,nlogn⟶n∞.(2)
FXsn′:nr1,nx+bn⟶nwpΦx+qI0,∞x if rj,nlogn⟶n∞, j=1,2, and r2,n/r1,n⟶n0, or if r1,nlogn⟶n∞ and r2,nlogn⟶nτ2≥0.(3)
FX,sn′:nr1,nx+bn⟶nwpΦx+qΦτx and rj,nlogn⟶n∞ if rj,nlogn⟶n⟶n∞, j=1,2, and r1,n/r2,n⟶nτ>0.Proof.
Under the conditionrj,nlogn⟶nτj≥0, j=1,2, the distribution of Xsn′:n can be written as(15)FXsn′:nanx+bn=PXsn′:n≤anx+bn=PUn+Vn≤x,where Un=Z0/anand Vn=Zsn′:n−bn/an are independent. From equation (5), Un=Ipr1,n/anY1,0+Iqr2,n/anY2,0, but rj,n/an=2rj,nlogn⟶n2τj, then(16)Un→np0,ifτ1=τ2=0,Iq2τ2Y2,0,ifτ1=0,τ2>0,Ip2τ1Y1,0,ifτ1>0,τ2=0,Ip2τ1Y1,0+Iq2τ2Y2,0,ifτ1,τ2>0.
Therefore, by using the law of total probability and some simple algebra, we get(17)PUn≤x⟶nwpI0,∞x+qΦx2τ2,ifτ1=0,τ2>0,pΦx2τ1+qI0,∞x,ifτ1>0,τ2=0,pΦx2τ1+qΦx2τ2,ifτ1,τ2>0.
Thus,(18)Un⟶np0,ifτ1=τ2=0,PUn≤x⟶nwpΩx,τ1+qΩx,τ2,ifmaxτ1,τ2>0.
On the other hand, we have(19)PVn≤x=PZsn′:n≤anx+bn,where Zsn′:n is the s th upper extreme OS based on the sequence Zii=1n, represented by (5), and Z1,Z2,…Zn are i.i.d RVs with common DF (6). Therefore,(20)ℱanx+bn=pΦanx+bn1−r1,n+qΦanx+bn1−r2,n.
Now, the asymptotic distribution ofZsn′:n can be found from Lemma 1 after determining the domain of attraction of the DF Φanx+bn/1−rj,n=Φaj,n∗x+bj,n∗aj,n∗=an/1−rj,n and bj,n∗=bn/1−rj,n. To do that, we use Khinchin’s type theorem and Extreme Value Theorem. First, we note that, from the assumption rj,nlogn⟶nτj (thus, rj,n⟶n0), we get aj,n∗/an=1/1−rj,n⟶n1. On the other hand, using the relations 1−rj,n−1/2=1+1/2rj,n1+O1and bn/an=2logn−1/2log logn+log4π and bearing in mind that log logn/logn⟶n0 (by using L’Hopital’s rule), we get bj,n∗−bn/an=an−1bn/1−rj,n−bn→nτj. Thus, Khinchin’s type theorem and Extreme Value theorem yield(21)Φnanx+bn1−rj,n⟶nwΨ3x+τj.
From (19)–(21) and Lemma 1, we get(22)PVn≤x⟶nwΨ3px+τ1Ψ3qx+τ2∑l=0s−1−plogΨ3x+τ1−qlogΨ3x+τ2ll!=exp−pux+τ1+qux+τ2∑l=0s−1pux+τ1+qux+τ2ll!=Hpux+τ1+qux+τ2.
Therefore, from (15) and Lemma 2.2.1 in [7], a combination of (18) and (22) yields the first part of the theorem. Now, turning to the case rj,nlogn⟶n∞, j=1,2, and r1,n/r2,n⟶n0, or r1,nlogn⟶nτ1≥0 and r2,nlogn⟶n∞. From representation (7), we get(23)PXsn′:n−bnr2,n≤x=PZ0r2,n+Zsn′:n−bnr2,n≤x.
From (5), we have Z0/r2,n=Ipr1,n/r2,nY1,0+IqY2,0→npIqY2,0 since r1,n/r2,n⟶n0. Therefore,(24)PZ0r2,n≤x⟶nwpI0,∞x+qΦx.
While, for everyε>0, we get(25)PZsn′:n−bnr2,n>ε=PZsn′:n−bnan>r2,nanε⟶n0,since r2,n/an=2r2,nlogn⟶n∞. Finally, from (23) and Lemma 2.2.1 in [7], a combination of (24) and (25) yields(26)PXsn′:n−bnr2,n≤x⟶nwpI0,∞x+qΦx.
In the same manner, under the conditionrj,nlogn⟶n∞, j=1,2, and r2,n/r1,n⟶n0, or r1,nlogn⟶n∞ and r2,nlogn⟶nτ2≥0, we get(27)PXsn′:n−bnr1,n≤x⟶nwpΦx+qI0,∞x,since Z0/r1,n⟶npIpY1,0 and pZsn′:n−bn/r1,n>ε⟶n0. Again, in the same manner, under the conditions rj,nlogn⟶n∞, j=1,2, and r1,n/r2,n⟶nτ>0, we get(28)PXsn′:n−bnr2,n≤x⟶nwpΦxτ+qΦx,since Z0/r2,n⟶nwIpτY1,0+IqY2,0 and pZsn′:n−bn/r2,n>ε⟶n0. Finally, we get(29)PXsn′:n−bnr1,n≤x⟶nwpΦx+qΦτx,since Z0/r1,n⟶nwIpY1,0+Iq1/τY2,0 and pZsn′:n−bn/r1,n>ε⟶n0. This completes the proof of the theorem.


## 3. Limit Distributions of Central OSs in a Mixture of Two SGSs

We will need two essential results (Lemmas2 and 3) due to [19, 20]. These two results will be formulated in a slightly different way that fits our study in this section, where the modifications made in these results are very easy to prove.Lemma 2 (see [20]).
Ifη1:n≤η2:n≤…≤ηn:n are the OSs corresponding to the i.i.d RVs η1,η2,…,ηn, with common DF Fη and probability density function (PDF) fη, such that F¯ηxλ≔1−Fηxλ=λ, where 0<λ<1 and fηxλ>0, then for any central rank sn′≔n−sn+1 satisfying the regular condition nsn/n−λ⟶n0, we have(30)Fηsn′:ncnx+xλ⟶nwΦx,where cn=λ1−λ/nfηxλ.Lemma 2 was originally given in [20] for ηsn:n with the same limit and the normalizing constants cn′=λ′1−λ′/nfηxλ′cn′=cn and xλ′xλ′=xλ, where λ′=1−λ and Fηxλ′=λ′.Lemma 3 (see [19]).
LetFi. be a given sequence of nondegenerate DFs and suppose η1,η2,…ηn are i.i.d RVs with common mixture DF:(31)Fηx=∑i=1kpiFiαi,nx,∑i=1kpi=1,αi,n>0.Furthermore, letηsn′:n≔ηn−sn+1:n (where nsn/n−λ⟶n0) be the sn′ th OS of the sequence η1,η2,…,ηn. Then, for some suitable linear transformation Gnx=anx+bn,an>0, we get(32)Fηsn′:nGnx⟶nwΦ∑i=1kpiuix;λ,ifFiαi,nGnx belongs to the central-domain of attraction of the nondegenerate type Φuix;λ, written Fiαi,nGnx∈DλΦuix;λ,i=1,2,…,k.It should be noted that ifuix;λ=ux;λ, for i=1,2,…,k, then Fηsn′:nGnx⟶nwΦux;λ.Remark 2.
In [19], Lemma 3 was proved for αi,n=1. In addition, according to the result of [20], there are only four possible limit types for Φuix;λ. Among these nondegenerate types, the normal type is an exclusively important type in our study due to Lemma 2.The following theorem gives the asymptotic distribution of central OSs concerning sequence (3) (and consequently sequence (7)).Theorem 2.
Concerning sequence (7), let Φxλ=1−λ and cn=λ1−λ/φxλn, where φx is the standard normal PDF. Then, for any central rank sn′≔n−sn+1 satisfying the regular condition nsn/n−λ⟶n0, we have(1)
FXsn′:ncnx+xλ→nwpΦ1+τ1φ2xλ/λ1−λ−1/2x+qΦ1+τ2φ2xλ/λ1−λ−1/2x, if nrj,n⟶nτj≥0, for j=1,2. Note that FXsn′:ncnx+xλ⟶nwΦx, if τ1=τ2=0.(2)
FXsn′:nr2,nx+xλ⟶nwpI0,∞x+qΦx, if nrj,n⟶n∞, j=1,2, and r1,n/r2,n⟶n0, or if nr1,n⟶nτ1≥0 and nr2,n⟶n∞.(3)
FXsn′:nr1,nx+xλ⟶nwpΦx+qI0,∞x, if nrj,n⟶n∞, j=1,2, and r2,n/r1,n⟶n0, or if nr1,n⟶n∞ and nr2,n⟶nτ2≥0.(4)
FXsn′:nr1,nx+xλ⟶nwpΦx+qΦτx and FXsn′:nr2,nx+xλ⟶nwpΦx/τ+qΦx, if nrj,n⟶n∞, j=1,2, and r1,n/r2,n⟶nτ>0.Proof.
Under the conditionnrj,n⟶nτj≥0,j=1,2, the distribution of Xsn′:n can be written as(33)FXsn′:ncnx+xλ=PXsn:n≤cnx+xλ=PUn+Vn≤x,where Un=Z0/cn and Vn=Zsn′:n−xλ/cn are independent. From equation (5), Un=Ipr1,n/cnY1,0+Iqr2,n/cnY2,0, but rj,n/cn=φxλnrj,n/λ1−λ→nφxλτj/λ1−λ, then(34)Un⟶npIpτ1φ2xλλ1−λY1,0+Iqτ2φ2xλλ1−λY2,0.
On the other hand, we have(35)PVn≤x=PZsn′:n≤cnx+xλ=FZsn′:ncnx+xλ,where Zsn′:n is the sn′ th central OS based on the sequence Zii=1n, represented by (5), and Z1,Z2,…Zn are i.i.d RVs with common DF (6). Therefore,(36)ℱcnx+xλ=pΦcnx+xλ1−r1,n+qΦcnx+xλ1−r2,n.
By relying on Khinchin’s type theorem and Lemma2, we are going to prove(37)Φcn1−rj,nx+xλ1−rj,n∈DλΦx,j=1,2.
Implication (37) holds since cj,n∗/cn=1/1−rj,n⟶n1 and bj,n∗−xλ/cn=xλ/cn1/1−rj,n−1→n0, where cj,n∗=cn/1−rj,n and bj,n∗=xλ/1−rj,n. The first limit is evident from the assumption nrj,n⟶nτj (thus,rj,n⟶n0). Hence, only the latter limit needs explanation. Using 1−rj,n−1/2=1+1/2rj,n1+°1 and rj,nn⟶n0, the result follows. Therefore, from (35)–(37) and Lemma 3, we get(38)PVn≤x⟶nwΦx.
Thus, from (33), a combination of (34) and (38) yields the first part of the theorem under the condition nrj,n⟶nτj≥0,j=1,2. Now, turn to the conditions nrj,n⟶n∞, j=1,2, and r1,n/r2,n⟶n0 or nr1,n⟶nτ1≥0 and nr2,n⟶n∞. From relation (7), we have(39)FXsn′:nr2,nx+xλ=PXsn′:n−xλr2,n≤x=PZ0r2,n+Zsn′:n−xλr2,n≤x.
From (5), we get Z0/r2,n=Ipr1,n/r2,nY1,0+IqY2,0→npIqY2,0 since r1,n/r2,n⟶n0. Therefore,(40)PZ0r2,n≤x⟶nwpI0,∞x+qΦx.
On the other hand, for everyε>0, we get(41)pZsn′:n−xλr2,n>ε=pZsn′:n−xλcn>r2,ncnε⟶n0,since r2,n/cn=φxλnr2,n/λ1−λ⟶n∞. Finally, from (39) and Lemma 2.2.1 in [7], a combination of (40) and (41) yields FXsn′:nr2,nx+xλ⟶nwpI0,∞x+qΦx. In the same manner under the conditions nrj,n⟶n∞,j=1,2, and r2,n/r1,n⟶n0 or nr1,n⟶n∞ and nr2,n⟶nτ2≥0, we get FXsn′:nr1,nx+xλ⟶nwpΦx+qI0,∞x, since Z0/r1,n→npIpY1,0 and pZsn′:n−xλ/r1,n>ε⟶n0, ∀ε>0. Also, under the condition nrj,n⟶n∞, j=1,2, and r1,n/r2,n⟶nτ>0, we get FXsn′:nr1,nx+xλ⟶nwpΦx+qΦτx, since Z0/r1,n→npIpY1,0+Iq1/τY2,0 and pXsn′:n−xλ/r1,n>ε⟶n0, ∀ε>0. Finally, FXsn′:nr2,nx+xλ⟶nwpΦx/τ+qΦx since Z0/r2,n→npIpτY1,0+IqY2,0 and pXsn′:n−xλ/r2,n>ε⟶n0, ∀ε>0.


## 4. Limit Distributions of Intermediate OSs in a Mixture of Two SGSs

Again in this section, we will need two essential results (Lemmas 4 and 5) due to [19, 21]. These two results will be formulated in a slightly different way that fits our study, where the modifications made in these results are very easy to prove.Lemma 4 (see [21]).
Ifη1:n≤η2:n≤…≤ηn:n are the OSs corresponding to the i.i.d RVs η1,η2,…,ηn, with common standard normal distribution Φx, then for any upper intermediate rank sn′≔n−sn+1 satisfying the condition sn/n⟶n0, we have(42)Φηsn′:nanx+bn⟶nwΦx,where an=sn/nϕbn∼1/bnsn, Φbn=1−sn/n, and bn∼2logn/sn, as n⟶∞.Lemma 5 (see [19]).
LetFi. be a given sequence of nondegenerate DFs and suppose η1,η2,…ηn are i.i.d RVs with common mixture DF,(43)Fηx=∑i=1kpiFiαi,nx,∑i=1kpi=1,αi,n>0.Furthermore, letηsn′:n≔ηn−sn+1:n (where sn/n⟶n0) be the sn′ th OS of the sequence η1,η2,…,ηn. Then, for some suitable Gnx=anx+bn,an>0, we get(44)Fηsn′:nGnx⟶nwΦ∑i=1kpiνix,if Fiαi,nGnx belongs to the intermediate-domain of attraction of the nondegenerate type Φνix, written Fiαi,nGnx∈DinΦνix,i=1,2,…,k. Note that if νix=νx, for i=1,2,…,k, then Fηsn′:nGnx⟶nwΦνx.Remark 3.
In [21], Lemma 4 was proved for ηsn:n (lower intermediate OSs). Moreover, in [19], Lemma 5 was proved for αi,n=1. In addition, according to the result of [22], there are only three possible limit types for Φνix. Among these nondegenerate types, the normal type is an exclusively important type in our study due to Lemma 4.The next theorem gives the limit DF of thesn′ th OS (upper intermediate OSs) of the sequence (7), under the condition logsn/logn⟶nβ, 0≤β≤1. It is worth mentioning that this condition is quite general, e.g., it is easily to check that it holds with β=α when sn∼lnα, 0<α<1 (the Chibisov rank sequence [19, 22]). Also, this condition will hold with β=0 when sn=logn. Finally, it will hold with β=1 when sn=n/logn.Theorem 3.
Concerning sequence (7), let logsn/logn⟶nβ, 0≤β≤1, an=sn/nϕbn∼1/bnsn, Φbn=1−sn/n, and bn∼2logn/sn, as n⟶∞, where φx be the standard normal PDF. Then, for any upper intermediate rank sn′, we have(1)
FXsn′:nanx+bn⟶nwpΦx/1+2τ11−β+qΦx/1+2τ21−β, if snrj,nlogn⟶nτj≥0, for j=1,2.(2)
FXsn′:nr2,nx+bn⟶nwpI0,∞x+qΦx, if snrj,nlogn⟶n∞, j=1,2, and r1,n/r2,n⟶n0, or if snr1,nlogn⟶nτ1≥0 and snr2,nlogn⟶n∞.(3)
FXsn′:nr1,nx+bn⟶nwpΦx+qI0,∞x, if snrj,nlogn⟶n∞,j=1,2, and r2,n/r1,n⟶n0, or if snr1,nlogn⟶n∞ and snr2,nlogn⟶nτ2≥0.(4)
FXsn′:nr1,nx+bn⟶nwpΦx+qΦτx and FXsn′:nr2,nx+bn⟶nwpΦx/τ+qΦx, if snrj,nlogn⟶n∞, j=1,2, and r1,n/r2,n⟶nτ>0.Proof.
Under the conditionsnrj,nlogn⟶nτj≥0, for j=1,2, the distribution of Xsn′:n concerning sequence (7) can be written as(45)FXsn′:nanx+bn=PXsn′:n≤anx+bn=PUn+Vn≤x,where Un=Z0/anand Vn=Zsn′:n−bn/an are independent. In view of (5), we get Un=Ipr1,n/anY1,0+Iqr2,n/anY2,0,but rj,n/an∼2rj,nsnlogn1−logsn/logn⟶n2τj1−β since an∼2snlogn/sn−1/2=2snlogn1−logsn/logn−1/2. Therefore,(46)Un→npIp2τ11−βY1,0+Iq2τ21−βY2,0.
Also,(47)PVn≤x=PZsn′:n≤anx+bn=FZsn′:nanx+bn,where Zsn′:n is the sn′ th intermediate OS based on the sequence Zii=1n, represented by (5), and Z1,Z2,…Zn are i.i.d RVs with common DF (6). Thus,(48)ℱanx+bn=pΦanx+bn1−r1,n+qΦanx+bn1−r2,n.
By using Khinchin’s type theorem and Lemma4, we are going to prove(49)Φan1−rj,nx+bn1−rj,n∈DinΦx,j=1,2.
Implication (49) holds, since aj,n∗/an=1/1−rj,n⟶n1 and bj,n∗−bn/an=bj,n∗−bn/an=bn/an1/1−rj,n−1→n0, where aj,n∗=an/1−rj,n and bj,n∗=bn/1−rj,n. The first limit is evident from the assumption snrj,nlogn⟶nτj (thus, rj,n⟶n0). Hence, only the latter limit needs explanation. Applying that 1−rj,n−1/2=1+1/2rj,n1+∘1, thus bj,n∗−bn/an=1/2bnrj,nrj,n/an1+∘1, but bnrj,n∼2rj,nlogn1−logsn/logn⟶n0 (from the assumption snrj,nlogn⟶nτj) and rj,n/an⟶n2τj1−β, which proves the second limit relation. Therefore, from (47)–(49) and Lemma 5, we get(50)PVn≤x⟶nwΦx.
Thus, from (45), a combination of (46) and (50) yields the first part of the theorem under the condition nrj,n⟶nτj≥0,j=1,2. Now, turning to the conditions snrj,nlogn⟶n∞, j=1,2, and r1,n/r2,n⟶n0, or snr1,nlogn→nτ1≥0 and snr2,nlogn⟶n∞. From relation (45), we have(51)FXsn′:nr2,nx+bn=PXsn′:n−bnr2,n≤x=PZ0r2,n+Zsn′:n−bnr2,n≤x.
From equation (5), we get Z0/r2,n=Ipr1,n/r2,nY1,0+IqY2,0→npIqY2,0 since r1,n/r2,n⟶n0. Therefore,(52)PZ0r2,n≤x⟶nwpI0,∞x+qΦx.
On the other hand, for everyε>0, we have(53)pZsn′:n−bnr2,n>ε=pZsn′:n−bnan>r2,nanε⟶n0,since r2,n/an∼2r2,nsnlogn1−logsn/logn⟶n∞. Thus, from (51) and Lemma 2.2.1 in [7], a combination of (52) and (53) yields(54)FXsn′:nr2,nx+bn⟶nwpI0,∞x+qΦx.
In the same manner, under the conditionssnrj,nlogn⟶n∞, j=1,2 and r2,n/r1,n⟶n0, or snr1,nlogn⟶n∞ and snr2,nlogn⟶nτ2≥0, we get FXsn′:nr1,nx+bn⟶nwpΦx+qI0,∞x since Z0/r1,n→npIpY1,0 and pZsn′:n−bn/r1,n>ε⟶n0, ∀ε>0. Also, under the conditions snrj,nlogn⟶n∞, j=1,2, and r1,n/r2,n⟶nτ>0, we get FXsn′:nr1,nx+bn⟶nwpΦx+qΦτx, since Z0/r1,n→npIpY1,0+Iq1/τY2,0 and pXsn′:n−bn/r1,n>ε⟶n0, ∀ε>0. Finally, FXsn′:nr2,nx+bn⟶nwpΦx/τ+qΦx, since Z0/r2,n→npIpτY1,0+IqY2,0 and pXsn′:n−bn/r2,n>ε⟶n0, ∀ε>0. This completes the proof of the theorem.


## 5. Limit Distributions of Record Values in a Mixture of Two SGSs

Chandler [23] published the key study on statistical treatment of the record values. He looked at the stochastic behavior of the random record values coming from i.i.d observations from a continuous DF F. The basic properties of the record values depend on the cumulative hazard function HFx=−logF¯x and its inverse Ψu=HF−1u=F−11−exp−u, e.g., the DF of the upper record value Rn based on the DF F may be expressed in terms of HFx as PRn≤x=ΓnHFx (cf. [15]), where Γnx=1/Γn∫0xtn−1e−tdt is the incomplete gamma ratio function. Resnick [24] derived the class of all possible limit laws of the upper record Rn. He connected between these limit laws and the max-limit laws through what is known as the Duality theorem via the associated DF Fax=1−exp−HFx. It is noteworthy that these limit laws are the same as the three possible limit laws for the intermediate OSs. For further interesting work related to the link between the OSs and record values, see [25]. Again, the normal type is an exclusively important type in our study due to the fact that the upper record based on the standard normal distribution belongs to the domain of attraction of the normal type DRΦ (cf. [15]). Namely,(55)PRn−Φ−11−e−nΦ−11−e−n+n−Φ−11−e−n≤x⟶nwΦx,where Φ−1x is the usual inverse function of Φx. Moreover, an equivalent simpler form of this limiting result is obtained by an application of the mean value theorem (cf. example 2.3.4 of [15]) as(56)PRn≤anx+bn⟶nwΦx,i.e., Φanx+bn∈DRΦx, where an=1/2 and bn=Φ−11−e−n.The following lemmas will be needed in our study. The first lemma (Lemma6) individually expresses an interesting fact concerning the mixture of normal DFs.Lemma 6.
Letm=minc,d>0. Then, 1−pΦcx−qΦdx∼m∗1−Φmx, as x⟶∞, where p=1−q and(57)m∗=p,ifm=c,q,ifm=d,1,ifm=c=d.Proof.
By using the obvious relations1−pΦcx−qΦdx⟶n0 and 1−Φmx⟶n0 and L’Hospital’s rule, we get (after some obvious algebra)(58)limx⟶∞1−pΦcx−qΦdx1−Φmx=limx⟶∞−pcφcx−qdφdx−mφmx=limx⟶∞pcexp−1/2x2c2−m2+qdexp−1/2x2d2−m2m=m∗,where φ is the PDF of the standard normal distribution.Lemma 7.
LetRnZ be the upper record value based on the DF ℱz, given by equation (6). Then,(59)PRnZ≤1−rnanx+bn⟶nwΦx,where rn=minr1,n,r2,n, 1≠maxr1,n,r2,n↛1, as n⟶∞, an=1/2 and bn=Φ−11−e−n.Proof.
Letun⟶n∞. Then, Lemma 6 yields(60)1−ℱun=1−pΦun1−r1,n−qΦun1−r2,n∼m∗1−Φun1−rn.
Takeun=1−rnanx+bn. Note that un⟶n∞, since bn∼2n⟶n∞ (cf. [10]). Now, we can easily get 1−ℱun∼m∗1−Φanx+bn. Thus, Hℱun∼−logm∗+HΦanx+bn. From relation (56), Corollary 6.4.1 in [7] (Tata [26] is credited with this corollary), and the last asymptotic relation, we get(61)limn⟶∞Hℱun−nn=limn⟶∞HΦanx+bn−nn=x.
Again, by Corollary 6.4.1 in [7], we get PRnZ≤1−rnanx+bn⟶nwΦx, as required to prove.The following theorem, which is the main result in this section, gives the asymptotic distribution of record values concerning sequence (8).Theorem 4.
LetRnX be the upper record value based on X1,X2,…Xn, given by equation (3). Let rn=minr1,n,r2,n, 1≠maxr1,n,r2,n↛1, as n⟶∞, an=1/2 and bn=Φ−11−e−n.(1)
Ifrj,nn⟶nτj≥0, for j=1,2, then  PRnX≤anx+bn⟶nwΦx+τ, where τ=τ1,ifrn=r1,n,τ2,ifrn=r2,n.(2)
Ifrj,nn⟶n∞, j=1,2, r1,n⟶n0, and r2,n⟶nr>0,      or if r1,nn⟶nτ≥0, r2,nn⟶n∞, and r2,n⟶nr>0, then PRnX≤1−r2,nanx+1−r1,nbn⟶nwpΦx+qΦx/2r+1.(3)
Ifrj,nn⟶n∞, j=1,2, r1,n⟶nr>0, and r2,n⟶n0,      or if r1,nn⟶n∞, r2,nn⟶nτ≥0, and r1,n⟶nr>0, then PRnX≤1−r2,nanx+1−r2,nbn⟶nwpΦx/2r+1+qΦx.(4)
Ifrj,nn⟶n∞, j=1,2, rj,n⟶nrj, and r1>0, then(62)PRnX≤r1,nx+1−rnbn⟶nwpΦ2r11+r1x+qΦ2r11+2r2−r1x,ifrn=r1,n,pΦ2r11+2r1−r2x+qΦ2r11+r2x,ifrn=r2,n.Proof.
Under the conditionrj,nn⟶nτj≥0, for j=1,2, by using the representation (8), the distribution of the record value RnX can be written as(63)PRnX≤anx+bn=PUn+Vn≤x,where Un=Z0/an and Vn=RnZ−bn/an are independent. Clearly, Un=Ipr1,n/anY1,0+Iqr2,n/anY2,0→np0, since rj,n⟶n0 from the assumption rj,nn⟶nτj≥0. On the other hand, by using Lemma 7 and Khinchin’s type theorem, we are going to prove the limit relation(64)PVn≤x=PRnZ≤anx+bn⟶nwΦx+τ.
Equation (64) holds, since An∗/An=1/1−rn⟶n1and Bn∗−Bn/An=bn/an1/1−rn−1⟶nτ, where An∗=an, Bn∗=bn, An=an1−rn, and Bn=bn1−rn. The first limit is evident from the assumption rj,nn⟶nτj≥0 (thus, rn⟶n0). Hence, only the second limit relation needs explanation. Since 1−rn−1/2=1+1/2rn1+∘1 and bn∼2n, Bn∗−Bn/An∼rnn1+∘1,as n⟶∞, which verifies the second limit relation. Now, from equations (63) and (64) and since Un→np0, we get PRnX≤anx+bn⟶nwΦx+τ. This completes the proof of the first part of the theorem. Turning now to prove the second part, i.e., we adopt the conditions rj,nn⟶n∞, j=1,2, r1,n⟶n0, and r2,n⟶nr, or r1,nn⟶nτ≥0, r2,nn⟶n∞, and r2,n⟶nr. First, we note that under the conditions of this case, we have minr1,n,r2,n=r1,n, for all large n. Thus, by using representation (8), the distribution of RnX can be written as(65)PRnX≤1−r1,nanx+1−r1,nbn=PUn+Vn≤x,where Un=Z0/1−r1,nanand Vn=RnZ−1−r1,nbn/1−r1,nan. Clearly, Un→npIq2rY2,0 and(66)PVn≤x=PRnZ≤1−r1,nanx+1−r1,nbn⟶nwΦx.
The remaining part of the proof is obvious. On the other hand, the proofs of the third and fourth parts of the theorem are identical to the proof of the second part, with the exception of the obvious changes, and we omit their proofs for brevity. The proof of the theorem is complete.Corollary 1.
Under the conditions of the first part of Theorem4 and by using Khinchin’s type theorem, we can easily prove the following two relations:(67)PRnX≤anx+1−rnbn⟶nwΦx,(68)PRnX≤1−rj,nanx+1−rnbn⟶nwΦx,j=1,2.Moreover, under the conditions of the second or third parts of Theorem4, if r=0, by using Khinchin’s type theorem, we can easily prove the following relation:(69)PRnX≤1−rnanx+1−rnbn⟶nwΦx.



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*Source: 7956195-2022-08-31.xml*