Records for Some Stationary Dependent Sequences
RRecords for Some Stationary Dependent Sequences
M. Falk, A. Khorrami and S. A. PadoanAugust 8, 2018
Abstract
For a zero-mean, unit-variance second-order stationary univariate Gaussian processwe derive the probability that a record at the time n , say X n , takes place and deriveits distribution function. We study the joint distribution of the arrival time process ofrecords and the distribution of the increments between the first and second record, andthe third and second record and we compute the expected number of records. We alsoconsider two consecutive and non-consecutive records, one at time j and one at time n and we derive the probability that the joint records ( X j , X n ) occur as well as theirdistribution function. The probability that the records X n and ( X j , X n ) take placeand the arrival time of the n -th record, are independent of the marginal distributionfunction, provided that it is continuous. These results actually hold for a second-orderstationary process with Gaussian copulas. We extend some of these results to the caseof a multivariate Gaussian process. Finally, for a strictly stationary process satisfyingsome mild conditions on the tail behavior of the common marginal distribution func-tion F and the long-range dependence of the extremes of the process, we derive theasymptotic probability that the record X n occurs and derive its distribution function.Keywords: Arrival time, closed skew-normal distribution, Gaussian process, general-ized extreme-value distribution, record, strictly stationary process. Let { X n , n ≥ } be a sequence of identically distributed random variables (rvs), and denoteby F the common univariate marginal distribution function. For any i, j ∈ N , set M i : j :=max( X i , . . . , X j ). For simplicity, we set M j := M j , that is M j := max( X , . . . , X j ). The1 a r X i v : . [ m a t h . S T ] A ug v X n is a record if X n > M n − . Such an event is coded by the indicator function R n := ( X n is a record). When X , X , . . . are independent, many results on records are alreadyknown (e.g., Galambos 1987; Arnold, Balakrishnan, and Nagaraja 1998; Resnick 2008, Ch.4; Barakat and Elgawad 2017; Falk et al. 2018). In the multivariate case various definitionsof records are possible and have been investigated both in the past and more recently, see e.g.,Goldie and Resnick (1989), Hashorva and H¨usler (2005), Hwang, Tsai, et al. (2010), Dombryet al. (2018) to name a few. In this work we consider complete records ; these are randomvectors which are univariate records in each component. Precisely, let { X n , n ≥ } be astrictly stationary sequence of d -dimensional random vectors (rvs) X n = ( X (1) n , . . . , X ( d ) n ) ∈ R d . Let F be the common joint distribution function of X n with margins F i , 1 ≤ i ≤ d .The rv X n is a complete record if X n > max ≤ i ≤ n − X i , where the maximum is computed componentwise. We denote the rv coding the occurrenceof a complete record at time n by R CR n := ( X n is a complete record).Except for Haiman (1987), Haiman et al. (1998), as far as we know, most of the availableresults on records concern sequences of independent random variables or vectors. In thepresent work we derive some new results on the records of a stationary sequence of depen-dent random variables and dependent random vectors, under appropriate conditions of thedependence structure.At first we consider a univariate second-order stationary Gaussian process with zero-mean, unit-variance. This means that for every n = 1 , , . . . , E( X n ) = 0, E ( X n ) = 1and the autocovariance of the process is translation-invariant depending only on the timedifference, i.e. for every i, j , ρ i,j = E( X i X j ) = E( X X j − i ) = ρ ,j − i ≡ ρ j − i , where ρ j − i is a function only of the separation j − i and for every m , ρ i + m,j + m = ρ j − i . We derivethe probability that a record at time n , say X n , takes place, and the distribution of X n ,being a record. Furthermore, we derive the joint distribution of the arrival time process ofrecords and more specifically the distribution of the increments between the first and secondrecord and the third and second record. We compute the expected number of records which,depending on the type of correlation structure of the Gaussian process, can be finite orinfinite. We also focus on joint records and we derive the probability that two consecutiveand non-consecutive records at the time j and n , say X j and X n , take place, as well as thejoint distribution of ( X j , X n ), considering they are both records.2e highlight that many of our findings, such as the probability that the records X n and ( X j , X n ) take place and the arrival time of the n -th record, are independent of themarginal distribution function F , provided that is is continuous. As a consequence, theresults actually hold for second-order stationary sequences with Gaussian copulas . On thecontrary the distribution of a record (two records), conditional to the assumption that it isa record (they are records), however does depend on F .Next we consider a strictly stationary process satisfying some mild conditions on the tailbehavior of the common marginal distribution function F and the long-range dependenceof the extremes of the process. More specifically, it is assumed that F is attracted by theso-called Generalized Extreme-Value family of distributions, and that maxima on separatedenough intervals within the time span n are approximately independent. Within this settingwe derive the probability that X n is a record, the distribution of X n (being a record), andthe expected number of records.We complete the work by considering a zero-mean, unit-variance multivariate second-order stationary Gaussian process. We derive the probability that a complete record at time n occurs, and we compute the distribution of X n (being a record), as well as the probabilitythat two complete records at the time j and n occur, and the joint distribution of ( X j , X n )(being records).The paper is organized as follows. In Section 2.1 we introduce some notation usedthroughout the paper and we briefly review some basic concepts on the multivariate closedskew-normal distribution. In Section 2.2 we present our main results on records for an uni-variate second-order stationary Gaussian process. In Section 2.3 we provide the asymptoticprobability and distribution function of a record at time n for a strictly stationary processthat satisfies some appropriate conditions. Finally, in Section 3 we extend some of the resultsderived in Section 2.3 to the case of multivariate second-order stationary Gaussian processes. Throughout the paper we use the following notation. The symbol X ∼ N n ( µ , Σ ), n ∈ N , means an n -dimensional random vector that follows a multivariate Gaussian distribu-tion with mean µ ∈ R n and positive-definite covariance matrix Σ = σ ¯ Σ σ ∈ R n,n , σ :=3iag( σ , . . . , σ nn ), and ¯ Σ is the correlation matrix. Its cumulative distribution function(cdf) and probability density function (pdf) are denoted by Φ n ( x ; µ , Σ ) and φ n ( x ; µ , Σ )with x ∈ R n . When µ = = (0 , . . . , (cid:62) and Σ = I , where I is the identity matrix, wewrite Φ n ( x ) for simplicity.We indicate with a,b ( a,b ) a matrix of dimension a × b whose elements are all equal toone (zero). We omit the subscripts when the dimensions of the matrices are clear from thecontext.We introduce the notion of a multivariate closed skew-normal (CSN) random vector andwe do so by using the so-called conditioning representation (Genton 2004, Ch. 2). Let U ∼ N m ( ξ , Ω ) being independent of V ∼ N n ( , Σ ), where ξ ∈ R m , Ω ∈ R m × R m and Σ ∈ R n × R n . Let ∆ ∈ R n × R m , then (cid:32) U ∆ U + V (cid:33) ∼ N m + n (cid:32)(cid:32) ξ (cid:33) , (cid:32) Ω Ω∆ (cid:62) ∆Ω Γ (cid:33)(cid:33) , where Γ = Σ + ∆Ω∆ (cid:62) . Define X equal to U , under the condition that ∆ U + V > µ ,denoted by X = ( U | ∆ U + V > µ ), where µ ∈ R n . The m -dimensional random vector X fol-lows a multivariate closed skew-normal distribution, in symbols X ∼ CSN m,n ( ξ , Ω , ∆ , µ , Σ ),whose pdf is, for all x ∈ R m , ψ m,n ( x ; ξ , Ω , ∆ , µ , Σ ) = φ m ( x − ξ ; Ω )Φ n ( ∆ ( x − ξ ); µ , Σ )Φ n ( ; µ , Γ ) . (1)We denote the cdf of X by Ψ m,n ( x ; ξ , Ω , ∆ , µ , Σ ). When ξ = , Ω = I and µ = , we omitthem among the parameters for simplicity and we write Ψ m,n ( x ; ∆ , Σ ) and ψ m,n ( x ; ∆ , Σ )instead. We recall that the closed skew-normal distribution is also known in the litera-ture as the unified multivariate skew-normal distribution, which simply uses a differentparametrization (e.g, Ch. 7.1.2 in Azzalini 2013). The exposition of our results benefitsfrom the parametrization used by the closed skew-normal distribution.We recall that if X ∼ CSN m,n ( ξ , Ω , ∆ , µ , Σ ) thenΨ m,n ( x ; ξ , Ω , ∆ , Σ ) = Φ n + m ( ˜ x ; ˜ Ω )Φ n ( ; µ , Γ ) , (2)where ˜ x = (cid:32) − µx − ξ (cid:33) , ˜ Ω = (cid:32) Γ − Ω∆ (cid:62) ∆Ω Ω (cid:33) , b ∈ R m and A ∈ R q,m then, b + X ∼ CSN m,n ( ξ + b , Ω , ∆ , µ , Σ ) (3) AX ∼ CSN q,n ( Aξ , Ω ∗ , ∆ ∗ , µ , Σ ∗ ) (4)where Ω ∗ = A Ω A (cid:62) , ∆ ∗ = ∆Ω A (cid:62) Ω ∗− and Σ ∗ = Γ − ∆ ∗ A Ω∆ (cid:62) , (see Ch. 2 in Genton2004 for details). Let { X n , n ≥ } be a second-order stationary Gaussian sequence of dependent rvs. Withoutloss of generality, assume for simplicity that E( X i ) = 0, E( X i ) = 1 for every 1 ≤ i ≤ n .Throughout the paper we will refer to such a process as a stationary standard Gaussian(SSG) sequence. For any n ∈ N , let I ⊂ { , . . . , n } and I (cid:123) = { , . . . , n } \ I identify the |I| -dimensional and |I (cid:123) | -dimensional subvector partition such that X = ( X , . . . , X n ) (cid:62) =( X (cid:62)I , X (cid:62)I (cid:123) ) (cid:62) , with corresponding partition of the parameter ¯ Σ . By | A | we denote the numberof elements of a set A .Our results rely on the following well-known important result on the conditional distribu-tion derived from joint Gaussian distribution. Precisely, let X = ( X (cid:62)I , X (cid:62)I (cid:123) ) (cid:62) ∼ N n ( µ , Σ )with corresponding partition of the parameters µ and Σ , then in Anderson (1984, Theorem2.5.1) it is established that the conditional distribution of X I (cid:123) given that X I = x I , is forall x I ∈ R |I| , X I (cid:123) | X I = x I ∼ N |I (cid:123) | (cid:0) µ I (cid:123) , Σ I (cid:123) , I (cid:123) ; I (cid:1) , µ I (cid:123) = Σ I (cid:123) , I ¯ Σ − I , I x I , Σ I (cid:123) , I (cid:123) ; I = ¯ Σ I (cid:123) , I (cid:123) − Σ I (cid:123) , I ¯ Σ − I , I Σ I , I (cid:123) . (5)Furthermore, we denote the related correlation matrix by¯ Σ I (cid:123) , I (cid:123) ; I = σ − I (cid:123) , I (cid:123) ; I Σ I (cid:123) , I (cid:123) ; I σ − I (cid:123) , I (cid:123) ; I , where σ I (cid:123) , I (cid:123) ; I = diag( Σ I (cid:123) , I (cid:123) ; I ). For any j ∈ { a, . . . , b } , when I = { j } we simplify thenotation writing X j and X a : b \ j = ( X a , . . . , X j − , X j +1 , . . . , X b ) (cid:62) . When j = a or j = b wefurther simplify the notation by X b = ( X , . . . X b ) (cid:62) and X b − = ( X , . . . , X b − ) (cid:62) .In our first result we compute the probability that X n is a record together with itsdistribution. It is well known that Pr( R n = 1) = 1 /n in the case of independent rv with5dentical continuous df (see e.g., Galambos 1987) and that the distribution of X n , given thatit is a record, equals that of the largest observation among X , . . . , X n (Falk et al. 2018). Proposition 2.1.
Let { X n , n ≥ } be a SSG sequence of rvs. For every n ≥ , let I = { n } , I (cid:123) = { , . . . , n − } . Then, the probability that X n is a record and the distribution of X n ,given that it is a record, are equal to Pr( R n = 1) = Φ n − ( ; Γ n − n − )Pr( X n ≤ x | R n = 1) = Ψ ,n − (cid:0) x ; (cid:37) n − , ¯ Σ n − , n − n (cid:1) , where Γ n − n − is a ( n − × ( n − variance-covariance matrix whose entries of theassociated correlation matrix ¯ Γ n − n − are γ i,j ; n = 1 + ρ i,j − ρ i,n − ρ j,n (cid:112) (1 − ρ i,n )(1 − ρ j,n ) , i (cid:54) = n, j (cid:54) = n (6) and ¯ Σ n − , n − n is a ( n − × ( n − correlation matrix with entries ρ i,j ; n = ρ i,j − ρ i,n ρ j,n (cid:113) (1 − ρ i,n )(1 − ρ j,n ) , i (cid:54) = n, j (cid:54) = n. Proof.
The probability that X n is a record isPr( X n > M n − ) = (cid:90) + ∞−∞ Pr (cid:16) X i < z, ∀ i ∈ I (cid:123) | X n = z (cid:17) φ ( z )d z = (cid:90) + ∞−∞ Pr( Z n − ≤ z (cid:37) n − ) φ ( z )d z = E Z { Pr( Z n − ≤ Z (cid:37) n − | Z ) } = Pr( Z n − − Z (cid:37) n − ≤ ) ≡ Φ n − ( ; Γ n − n − ) , where Γ n − n − = ¯ Σ n − , n − n + (cid:37) n − (cid:37) (cid:62) n − (7) (cid:37) n − = σ − n − , n − n ( n − − ¯ Σ n − ,n ) = (cid:32)(cid:115) − ρ i,n ρ i,n , ∀ i ∈ I (cid:123) (cid:33) (cid:62) . (8)To obtain the second line we used the formula in (5), which leads to Z n − = σ − n − , n − n ( X n − − µ n ) ∼ N n − ( ; ¯ Σ n − , n − n ), where µ n = ( ρ i,n , ∀ i ∈ I (cid:123) ) (cid:62) v , and this can be seen6s independent of Z ∼ N (0 , X n ,Pr( X n ≤ x | R n = 1) = Pr( X n ≤ x, X n > M n − )Pr( X n > M n − ) , = (cid:82) x −∞ φ ( z )Φ n − ( z (cid:37) n − ; ¯ Σ n − , n − n )d z Φ n − ( ; Γ n − n − ) ≡ Ψ ,n − (cid:0) x ; (cid:37) n − , ¯ Σ n − , n − n (cid:1) . The correlations ρ i,j , 1 ≤ i < j ≤ n , in Proposition 3.1 satisfy − ≤ ρ i,j ; n ≤ − ≤ γ i,j ; n ≤ ρ i,n + ρ j,n − − (cid:113) (1 − ρ i,n )(1 − ρ j,n ) ≤ ρ i,j ≤ ( ρ i,n + ρ j,n −
1) + 2 (cid:113) (1 − ρ i,n )(1 − ρ j,n ) . Remark . Assume in Proposition 3.1 that ρ i,j = 0 for all ≤ i (cid:54) = j ≤ n . Then, Pr ( R n = 1) = Φ n − ( ; I n − + n − (cid:62) n − )= E (Φ n − ( n − Z ; I n − ))= (cid:90) + ∞−∞ Φ n − ( n − z ; I n − ) φ ( z ) d z = (cid:90) + ∞−∞ Φ n − ( z ) φ ( z ) d z = n − , where Z ∼ N (0 , . As expected, we obtain the results in (Galambos 1987) and Lemma 1.1in (Falk, Chokami, and Padoan 2018). Furthermore, Pr( X n ≤ x | R n = 1) = Ψ ,n − ( x ; n − , I n − )= n (cid:90) x −∞ Φ n − ( n − z ; I n − ) φ ( z ) d z = n (cid:90) x −∞ Φ n − ( z ) φ ( z ) d z = Φ( x ) n . Let T ( k ) := inf (cid:40) m ∈ N : m (cid:88) i =1 R i = k (cid:41) , k ≥ , T (1) := 1 , be the arrival time of the k -th record. Lemma 2.3.
Let { T ( k ) } k ≥ be the arrival time process of records. Let I = { j , . . . , j k } where ≤ j < · · · < j k ∈ N and j := 1 . Set I (cid:123) := { , . . . , j k } \ I . Then, Pr( T ( i ) = j i , i = 2 , . . . , k )= Φ j k − k ( ; Γ I (cid:123) , I (cid:123) )Ψ k − ,j k − k ( ; D ¯ Σ I , I D (cid:62) , ∆ , ¯ Σ I (cid:123) , I (cid:123) ; I )7 here D = ( I k − k − ) − ( k − I k − ) , ∆ = (cid:37) I (cid:123) , I (cid:123) ¯ Σ I , I D (cid:62) ( D ¯ Σ I , I D (cid:62) ) − , (9) Γ I (cid:123) , I (cid:123) = (cid:37) I (cid:123) , I (cid:123) ¯ Σ I , I (cid:37) (cid:62)I (cid:123) , I (cid:123) + ¯ Σ I (cid:123) , I (cid:123) ; I , (10) (cid:37) I (cid:123) , I (cid:123) = σ − I (cid:123) , I (cid:123) ; I ( B − Σ I (cid:123) , I ¯ Σ − I , I ) , (11) and B := j − j − . . . j − j − j − j − j − j − . . . j − j − j − j − ... ... ... ... j k − j k − − j k − j k − − . . . j k − j k − − j k − j k − − ∈ R j k − k,k − (12) Proof.
We havePr( T ( i ) = j i , i = 2 , . . . , k )= Pr( M j i +1: j i +1 − < X i , i = 1 , . . . , k − , X j k − < X j k )= (cid:90) + ∞−∞ (cid:90) z k −∞ · · · (cid:90) z −∞ Pr( M j i +1: j i +1 − < z i , i = 1 , . . . , k − | X j i = z i , i = 1 , . . . , k − · φ k ( z , . . . , z k )d z . . . d z k = (cid:90) + ∞−∞ (cid:90) z k −∞ · · · (cid:90) z −∞ Pr( X I (cid:123) < Bz | X I = z ) φ k ( z ; ¯ Σ I , I )d z where B is given in (12). By standardizing the random vector X I (cid:123) , we obtain (cid:90) + ∞−∞ (cid:90) z k −∞ · · · (cid:90) z −∞ Φ j k − k ( (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) , I (cid:123) ; I ) φ k ( z ; ¯ Σ I , I )d z = Φ j k − k ( ; Γ I (cid:123) , I (cid:123) ) (cid:90) + ∞−∞ (cid:90) z k −∞ · · · (cid:90) z −∞ ψ k,j k − k ( ; ¯ Σ I , I , (cid:37) I (cid:123) , I (cid:123) , ¯ Σ X I (cid:123) , X I (cid:123) ; X I )d z = Φ j k − k ( ; Γ I (cid:123) , I (cid:123) ) Pr( Z < Z < · · · < Z k )= Φ j k − k ( ; Γ I (cid:123) , I (cid:123) ) Pr( Z − Z < , . . . , Z k − − Z k < Γ I (cid:123) , I (cid:123) and (cid:37) I (cid:123) , I (cid:123) are given in (10) and (11).By recalling formula (4), we obtain Z − Z ...Z k − − Z k = − . . .
00 1 − . . . . . . . . . − Z ...Z k = DZ ∼ CSN k − ,j k − k ( D ¯ Σ I , I D (cid:62) , ∆ , ¯ Σ I (cid:123) , I (cid:123) ; I )8here ∆ is given in (9)In the next result we establish the distribution of the arrival time T (2) of the secondrecord as well as that of the increment X T (2) − X . Theorem 2.4.
Let { X n , n ≥ } be a SSG sequence of rvs. Let ρ i,j = E( X i , X j ) with ≤ i (cid:54) = j ≤ n . Assume that for n → ∞ , ρ i,j → as | j − i | → ∞ and ρ k,n → as k → ∞ .For n = 2 , , . . . , the distribution of the arrival time of the second record T (2) is Pr ( T (2) = n ) = / , n = 2 , Φ n − ( ; Γ n − , n − ) − Φ n − ( ; Γ n, n ) , n > where Γ n − , n − and Γ n, n are defined similarly to (7) . Furthermore, for every x > , thedistribution of the increment X T (2) − X is H ( x ) = (cid:88) n ≥ Φ n − ( u x ; Γ n, n ) − Φ n − ( ; Γ n, n ) , (14) where u x = ( x/ (1 − ρ ,n ) / , , . . . , (cid:62) is an ( n − -dimensional vector.Proof. When n = 2 we havePr( T (2) = 2) = Pr( X > X ) = 1 / . For n > T (2) = n ) = Pr ( X i < X , i = 2 , . . . , n − , X n > X )= Pr ( X i < X , i = 2 , . . . , n − − Pr ( X i < X , i = 2 , . . . , n ) . Therefore, (13) follows by similar arguments to those used in Proposition 3.1. It must bechecked that (cid:88) n ≥ Pr( T (2) = n ) = 12 + lim N →∞ N (cid:88) n =3 (Φ n − ( ; Γ n − , n − ) − Φ n − ( ; Γ n, n ))= lim N →∞ (1 − Φ ( ; Γ , ) + Φ ( ; Γ , ) − · · · + Φ N − ( ; Γ N − , N − ) − Φ N − ( ; Γ N − , N − ))= 1 − lim N →∞ Φ N − ( ; Γ N − , N − ) = 1 . Let ( ˜ X , . . . , ˜ X n − ) be zero-mean unit-variance Gaussian sequence with variance-covariancematrix Γ n − , n − . Set P n = Pr( ˜ X i ≤ , . . . , ˜ X n − ≤
0) = Φ n − ( ; Γ n − , n − ). Clearly9 n − ( ; Γ n − , n − ) = Φ n − (cid:0) ; ¯ Γ n − , n − (cid:1) . We recall that Pr( ˜ X i ≤
0) = 1 / i = 1 , . . . , n −
1. By the Fr´echet inequalities we have that A n := max (cid:32) , n (cid:88) i =1 Pr( X i ≤ − ( n − (cid:33) = max(0 , − n/ ≤ P n ≤ / . For P n we derive the following upper bound B n . Precisely, P n = Pr (cid:32) n − (cid:88) i =1 ( ˜ X i ≤ ≥ n − (cid:33) = Pr (cid:40) n − (cid:88) i =1 (cid:18) ( ˜ X i ≤ − (cid:19) ≥ n − (cid:41) ≤ Pr (cid:40)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:88) i =1 (cid:18) ( ˜ X i ≤ − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ n − (cid:41) ≤ n − E (cid:40) n − (cid:88) i =1 (cid:18) ( ˜ X i ≤ − (cid:19)(cid:41) = 4( n − n − (cid:88) i =1 n − (cid:88) j =1 Cov( ( ˜ X i ≤ , ( ˜ X j ≤ n − n − (cid:88) i =1 n − (cid:88) j =1 Cov( P i,j ; n − /
4) =: B n , where P i,j ; n := Pr( ˜ X i ≤ , ˜ X j ≤
0) = Φ (0; γ i,j ; n ) and where Φ ( · ; γ i,j ; n ) is a bivariateGaussian cdf with correlation γ i,j ; n that is given in (6). In the third row we used theChebyshev’s inequality. Set h = | j − i | we rewrite B n as B n = 4( n − n − (cid:88) h =0 n − h )( P h ; n − / n (1 + 2 /n ) ( P n − /
4) + 8 n (1 + 2 /n ) n − (cid:88) h =1 (cid:18) − hn (cid:19) ( P h ; n − / α n + β n , where P h ; n := Pr( ˜ X ≤ , ˜ X h ≤
0) = Φ (0; γ h ; n ) and γ h ; n = 1 + ρ ,h − ρ ,n − i − ρ h,n − i (cid:112) (1 − ρ ,n − i )(1 − ρ h,n − i ) , h = 0 , . . . , n − . Now, when h = 0 we obtain γ n = 1 and therefore P n = 1 / α n → n → ∞ . We rewrite the term β n as β n = 8 n (1 + 2 /n ) n − (cid:88) h =1 ( P h ; n − / − n (1 + 2 /n ) n − (cid:88) h =1 hn ( P h ; n − / c n − d n . Now by the assumption we have that for n → ∞ , γ h ; n → h → ∞ , therefore for all ε > n such that for all h > n we have | P h ; n − / | < ε . As a consequence we have c n = 8 n (1 + 2 /n ) (cid:32) n (cid:88) h =1 ( P h ; n − /
4) + n − (cid:88) h = n +1 ( P h ; n − / (cid:33) < n (1 + 2 /n ) ( c + ε ( n − n + 1)) = o (1) , where c is a positive constant. Therefore, c n → n → ∞ and since d n < c n then β n → B n → n → ∞ . Concluding, since A n ≤ P n ≤ B n and A n = 0 for n ≥
2, then P n → n → ∞ .Finally, for every x > X T (2) − X isPr( X T (2) − X ≤ x ) = (cid:88) n ≥ Pr( X n − X ≤ x, T (2) = n )= (cid:88) n ≥ Pr( X n − X ≤ x, X i < X , i = 2 , . . . , n − , X n > X )= (cid:88) n ≥ Pr(0 < X n − X ≤ x, X i < X , i = 2 , . . . , n − < X n − X ≤ x, X i < X , i = 2 , . . . , n − (cid:90) + ∞−∞ Pr(0 < X n − u ≤ x, X i < u, i = 2 , . . . , n − | X = z ) φ ( z )d z = (cid:90) + ∞−∞ Pr( X i < x, i = 2 , . . . , n − , X n ≤ z + x | X = z ) φ ( z )d z − (cid:90) + ∞−∞ Pr( X i < z, i = 2 , . . . , n | X = z ) φ ( z )d z. Therefore, (14) follows by similar arguments to those used in Proposition 3.1.
Remark . Note that when ρ i,j = 0 for all ≤ i < j ≤ n and n > we obtain Pr ( T (2) = n ) = Φ n − ( ; I n − + n − (cid:62) n − ) − Φ n − ( ; I n − + n − (cid:62) n − ) = 1 n − − n = 1 n ( n − . Let N := (cid:80) ∞ n =1 R n be the number of records among an infinite sequence X , X , . . . When the components of the sequence are independent and identically distributed with a11ontinuous df, then it is a well-known result that an infinite number of records will occur: E ( N ) = (cid:80) ∞ n =1 P ( R n = 1) = (cid:80) ∞ n =1 /n = ∞ (Galambos 1987).A natural question that arises is the following. What is the expected number of recordsthat will take place in the case of a stationary Gaussian process? Proposition 2.6.
Let { X n , n ≥ } be a SSG sequence of rvs and Φ n − ( ; Γ n − n − ) bethe probability that a record take place described in Proposition 3.1. Let N be the number ofrecords among an infinite sequence X , X , . . . Then, we have E( N ) = ∞ , if / ≤ γ i,j ; n ≤ , ∀ ≤ i (cid:54) = j < n , if γ i,j ; n = 0 , ∀ ≤ i (cid:54) = j < n. where γ i,j ; n is the correlation parameter in (6) .Proof. First, note that E( N ) = E (cid:32) ∞ (cid:88) n =1 R n (cid:33) = ∞ (cid:88) n =1 E( R n )= 1 + ∞ (cid:88) n =2 Pr( X n > M n − )= 1 + ∞ (cid:88) n =2 Φ n − ( ; Γ n − , n − ) . The entries of the correlation matrix ¯ Γ n − , n − in (6) are γ i,j ; n = 1 /
2, 1 ≤ i (cid:54) = j < n , ifand only if ρ i,j = ρ i,n = ρ j,n = 0. In this case by Remark 2.2 we have that Φ n − ( ; I n − + n − (cid:62) n − ) = 1 /n . From this it follows that when 1 / ≤ γ i,j ; n ≤ (cid:113) (1 − ρ i,n )(1 − ρ j,n ) ≤ ρ i,j − ρ i,n − ρ j,n ≤ (cid:113) (1 − ρ i,n )(1 − ρ j,n ) , (15)then Φ n − ( ; Γ n − , n − ) ≥ /n and as a consequenceE( N ) ≥ ∞ (cid:88) n =1 n = ∞ . For every 1 ≤ i (cid:54) = j < n , provided that ρ i,n + ρ j,n ≥
0, when ρ i,j = ρ i,n + ρ j,n − Γ n − , n − = I n − . Therefore in this case Φ n − ( ; Γ n − , n − ) = Φ n − ( I n − ) = 2 − n +1 .As a consequence E( N ) = 1 + ∞ (cid:88) n =2 − n +1 = 2 ∞ (cid:88) n =0 − n − . X i and X j are more correlated thanthe sum of the correlations between X i and X n , and X j and X n , for every 1 ≤ i (cid:54) = j < n .The second assertion follows from the left-hand side of the inequality in (15) by noting that0 ≤ (cid:112) (1 − ρ i,n )(1 − ρ j,n ) ≤
1. This suggests looking at 1+ ρ i,j − ρ i,n − ρ j,n ≥ ρ i,j ≥ ρ i,n + ρ j,n . Instead, loosely speaking when X i and X j are less correlated thanthe sum of the correlations between X i and X n , and X j and X n , for every 1 ≤ i (cid:54) = j < n ,the expected number of records can be finite. This assertion follows from the condition ρ i,j = ρ i,n + ρ j,n −
1, provided that ρ i,n + ρ j,n ≥
0, which leads that two records should beexpected.In our next result we compute the distribution of the interarrival time between the secondand third record.
Proposition 2.7.
The distribution of the increment has the representation
Pr( X T (3) − X T (2) ≤ x )= ∞ (cid:88) j =2 ∞ (cid:88) k = j +1 Φ k − ( ; Γ I (cid:123) , I (cid:123) ) (cid:8) Ψ ,k − ( ; D ¯ Σ I , I D (cid:62) , ∆ , ¯ Σ I (cid:123) , I (cid:123) ; I ) − Ψ ,k − ((0 , − x ); D ¯ Σ I , I D (cid:62) , ∆ , ¯ Σ I (cid:123) , I (cid:123) ; I ) (cid:9) , (16) where the sets of indices I = { , j, k } and I (cid:123) = { , . . . , j − , j + 1 , . . . , k − } vary with j and k , ∆ and ˜ (cid:37) I (cid:123) , I (cid:123) are similarly defined as in formula (9) and (11) and where D := (cid:32) − − (cid:33) Proof.
By the total probability rulePr( X T (3) − X T (2) ≤ x ) = ∞ (cid:88) j =2 ∞ (cid:88) k = j +1 Pr( X k − X j ≤ x, T (3) = k, T (2) = j )13ote that, by repeating the same arguments as the previous proofsPr( X k − X j ≤ x, T (3) = k, T (2) = j )= Pr( X k − X j ≤ x, M j − < X , X < X j , M j +1: k − < X j , X j < X k )= (cid:90) + ∞−∞ (cid:90) z k z k − x (cid:90) z j −∞ Pr( M j − < z , M j +1: k − < z j ) φ ( z , z j , z k )d z d z j d z k = (cid:90) + ∞−∞ (cid:90) z k z k − x (cid:90) z j −∞ Φ k − (˜ (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) , I (cid:123) ; I ) φ ( z ; ¯ Σ I , I )d z = Φ k − ( ; Γ I (cid:123) , I (cid:123) ) Pr( Z < Z j < Z k ) − Φ k − ( ; Γ I (cid:123) , I (cid:123) ) Pr( Z < Z j < Z k − x ) , and thus, the assertion follows by repeating the arguments in the proof of Lemma 2.3.In the following result we derive the probability that two records occur at prescribedindices, with no further record in between, together with the distribution of such consecutiverecords. Theorem 2.8.
Let { X n , n ≥ } be a SSG sequence of rvs. For every n ≥ and j < n , let I = { j, n } , I (cid:123) = { , . . . , j − , j + 1 , . . . , n − } . The probability that two consecutive records X j and X n occur, is Pr (cid:0) R j = 1 , R n = 1 , ∩ n − i = j +1 R i = 0 (cid:1) = Φ n − ( ; Γ n − \ j, n − \ j ) − Φ n − ( ; Γ n \ j, n \ j ) , (17) where Γ n − \ j, n − \ j and Γ n \ j, n \ j are similarly defined as in (7) . The joint distribution of ( X j , X n ) , given that they are consecutive records, is Pr (cid:0) X j ≤ x , X n ≤ x | R j = 1 , R n = 1 , ∩ n − i = j +1 R i = 0 (cid:1) = P ( x , x ) , x ≤ x P ( x , x ) x > x where P ( a, b ) = w n − ( b µ ; ˜ Γ n \ j, n \ j )Ψ ,n − (cid:0) a ; ˜ (cid:37) n \ j , − b µ , ¯ Σ n \ j, n \ j ; j (cid:1) − w n − ( ; Γ n \ j, n \ j )Ψ ,n − (cid:0) a ; (cid:37) n \ j , , ¯ Σ n \ j, n \ j ; j (cid:1) and where (cid:37) n \ j is similarly defined as in (8) , ˜ Γ n \ j, n \ j = ¯ Σ n \ j, n \ j ; j + ˜ (cid:37) n \ j ˜ (cid:37) (cid:62) n \ j with ˜ (cid:37) n \ j = (cid:37) (cid:62) n \ j , − ρ n,j (cid:113) − ρ n,j (cid:62) , = (cid:16) , . . . , , (1 − ρ j,n ) − / (cid:17) (cid:62) ∈ R n − . and for any x ∈ R n − and positive-definite matrix Σ ∈ R n − ,n − , w n − ( x ; Σ ) = Φ n − ( x ; Σ )Φ n − ( ; Γ n − \ j, n − \ j ) − Φ n − ( ; Γ n \ j, n \ j ) . Proof.
First we compute the probability that two consecutive records occur. For every1 ≤ j < n we havePr( R j = 1 , R n = 1 , ∩ n − i = j +1 R i = 0) = Pr( X j > M j − , X n > M n − )= Pr( X i < X j , ∀ i ∈ I (cid:123) , X n > X j )= Pr( X i < X j , ∀ i ∈ I (cid:123) , ) − Pr( X i < X j , ∀ i ∈ { i, . . . , n } \ { j } ) . Therefore, (17) follows by similar arguments to those used in Proposition 3.1.The joint distribution of ( X j , X n ) is given byPr (cid:0) X j ≤ x , X n ≤ x | R j = 1 , R n = 1 , ∩ n − i = j +1 R i = 0 (cid:1) = Pr (cid:0) X j ≤ x , X n ≤ x , X j > M j − , X n > M n − , ∩ n − i = j +1 R i = 0 (cid:1) Pr( X j > M j − , X n > M n − ) . Note that Pr( X j ≤ x , X n ≤ x , X j > M j − , X n > M n − , ∩ n − i = j +1 R i = 0)= Pr( X j ≤ x , X n ≤ x , X i < X j , ∀ i ∈ I (cid:123) , X j < X n )= Pr( X j ≤ x , X n ≤ x , X i < X j , ∀ i ∈ I (cid:123) ) − Pr( X j ≤ x , X n ≤ x , X i < X j , i = 1 , . . . , n, i (cid:54) = j )= A ( x , x ) − B ( x , x ) . When x ≤ x , we obtain from similar arguments as those used in the proof of Proposition3.1 A ( x , x ) := Pr( X j ≤ x , X n ≤ x , X i < X j , ∀ i ∈ I (cid:123) )= (cid:90) x −∞ Pr( X n ≤ x , X i < z, ∀ i ∈ I (cid:123) | X j = z ) φ ( z )d z = (cid:90) x −∞ Pr Z i < (cid:115) − ρ i,j ρ i,j z, ∀ i ∈ I (cid:123) , Z n < − ρ n,j (cid:113) − ρ n,j z + x (cid:113) − ρ n,j φ ( z )d z = (cid:90) x −∞ Φ n − (˜ (cid:37) n \ j z + x µ ; ¯ Σ n \ j, n \ j ; j ) φ ( z )d z = Φ n − ( x µ ; ˜ Γ n \ j, n \ j )Ψ ,n − (cid:0) x ; ˜ (cid:37) n \ j , − x µ , ¯ Σ n \ j, n \ j ; j (cid:1) . B ( x , x ) := Pr( X j ≤ x , X i < X j , i = 1 , . . . , n, i (cid:54) = j )= (cid:90) x −∞ Pr( X i < z, i = 1 , . . . , n, i (cid:54) = j | X j = z ) φ ( z )d z = Φ n − ( ; Γ n \ j, n \ j )Ψ ,n − (cid:0) x ; (cid:37) n \ j , , ¯ Σ n \ j, n \ j ; j (cid:1) . When x > x , it is sufficient to compute A ( x , x ) and B ( x , x ) in ( x , x ).In the next result we drop the assumption that the two records in Theorem 2.8 areconsecutive. Theorem 2.9.
Let { X n , n ≥ } be a SSG sequence of rvs. For every n ≥ and j < n , let I = { j, n } and I (cid:123) = { , . . . , j − , j + 1 , . . . , n − } . The probability that X j and X n arerecords, is Pr( R j = 1 , R n = 1) = Φ n − ( ; ˜ Ω ) . The joint distribution of ( X j , X n ) , given that they are records, is Pr( X j ≤ x , X n ≤ x | R j = 1 , R n = 1) = P ( x , x ) , x ≤ x P ( x , x ) x > x where P ( a, b ) = Φ n − ( ; Γ I (cid:123) , I (cid:123) )Φ n − ( ; ˜ Ω ) (cid:16) Ψ ,n − ( a, b ; ¯ Σ I , I , (cid:37) I (cid:123) , I (cid:123) , ¯ Σ I (cid:123) , I (cid:123) ; I ) − Ψ ,n − ( a, a ; ¯ Σ I , I , (cid:37) I (cid:123) , I (cid:123) , ¯ Σ I (cid:123) , I (cid:123) ; I )+ Ψ ,n − (0 , a ; D ¯ Σ I , I D (cid:62) , ∆ , ¯ Σ ∗∗I (cid:123) , I (cid:123) ; I ) (cid:17) . Proof.
Similar steps as those used in the proof of Proposition 3.1 show that the probabilitythat X j and X j are records, isPr( R j = 1 , R n = 1) = Pr( X j > M j − , X n > M n − )= (cid:90) + ∞−∞ (cid:90) z −∞ Pr (cid:0) M j − < z , M n − j +1 < z | X j = z , X n = z (cid:1) φ ( z , z ; ¯ Σ I , I )d z d z = (cid:90) + ∞−∞ (cid:90) z −∞ Φ n − (cid:0) (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) , I (cid:123) ; I (cid:1) φ ( z ; ¯ Σ I , I )d z d z = Φ n − ( , Γ I , I (cid:123) ) (cid:90) + ∞−∞ (cid:90) z −∞ ψ ,n − (cid:0) z ; ¯ Σ I , I , (cid:37) I (cid:123) , I (cid:123) , ¯ Σ I (cid:123) , I (cid:123) ; I (cid:1) d z d z = Φ n − ( ; Γ I , I (cid:123) ) Pr( Z − Z < , Z , Z ) ∼ CSN ,n − ( ¯ Σ I , I , (cid:37) I (cid:123) , I (cid:123) , ¯ Σ I (cid:123) , I (cid:123) ; I ). Precisely, to obtain the third line we usedthe formula in (5) and where B = (cid:32) j − j − n − j − n − j − (cid:33) , (cid:37) I (cid:123) , I (cid:123) = σ − I (cid:123) , I (cid:123) ; I ( B − Σ I (cid:123) , I ¯ Σ − I , I )= σ (cid:16) − ρ j − ρ n ρ jn − ρ jn (cid:17) ρ j ρ jn − ρ n σ (1 − ρ jn ) ... ... σ j − ,j − (cid:16) − ρ j − ,j − ρ j − ,n ρ jn − ρ jn (cid:17) ρ j − ,j ρ jn − ρ j − ,n σ j − ,j − (1 − ρ jn ) ρ j +1 ,j ρ jn − ρ j +1 ,n σ j +1 ,j +1 (1 − ρ jn ) 1 σ j +1 ,j +1 (cid:16) − ρ j +1 ,j − ρ j +1 ,n ρ jn − ρ jn (cid:17) ... ... ρ n − ,j ρ jn − ρ n − ,n σ n − ,n − (1 − ρ jn ) 1 σ n − ,n − (cid:16) − ρ n − ,j − ρ n − ,n ρ jn − ρ jn (cid:17) (18)and ¯ Σ I (cid:123) , I (cid:123) ; I is a ( n − × ( n −
2) partial correlation matrix with upper diagonal entries ρ i,k ; j,n = ρ ij − ρ ij − ρ in ρ jn − ρ jn ρ kj − ρ in − ρ ij ρ jn − ρ jn ρ kn , ∀ i < k ∈ I (cid:123) . and σ i,i = 1 − ρ ij − ρ in ρ jn − ρ jn ρ ij − ρ in − ρ ij ρ jn − ρ jn ρ in . In the third line we multiply and divide the term within the integrals with Φ n − ( ; Γ I (cid:123) , I (cid:123) )where Γ I (cid:123) , I (cid:123) is defined as Γ I (cid:123) , I (cid:123) = (cid:37) I (cid:123) , I (cid:123) ¯ Σ I , I (cid:37) (cid:62)I (cid:123) , I (cid:123) + ¯ Σ I (cid:123) , I (cid:123) ; I . We therefore recognize a unified multivariate skew-normal pdf within the integrals and theintegral of it can be seen as Pr( Z < Z ). Now, by (4) we obtain Z − Z = (cid:16) − (cid:17) (cid:32) Z Z (cid:33) ∼ CSN ,n − (cid:16) − ρ j,n ) , ∆ ∗ , Σ ∗I (cid:123) , I (cid:123) ; I (cid:17) where ∆ ∗ = 12(1 − ρ j,n ) (cid:37) I (cid:123) , I (cid:123) ¯ Σ I , I (cid:32) − (cid:33) = 12 (cid:32) j − − n − j − (cid:33) (cid:18) σ ii (cid:18) ρ in − ρ ij − ρ jn (cid:19)(cid:19) i =1 ,...,n − ,i (cid:54) = j Σ ∗I (cid:123) , I (cid:123) ; I = Γ I (cid:123) , I (cid:123) − ∆ ∗ (cid:16) − ρ jn − ρ jn (cid:17) (cid:37) (cid:62)I (cid:123) , I (cid:123) . By formula (2) we obtain the result, with˜ Ω = (cid:32) Γ I (cid:123) , I (cid:123) − ρ jn ) ∆ ∗(cid:62) − ρ jn ) ∆ ∗ − ρ jn ) (cid:33) . By similar steps we can compute the joint distribution of two records ( X j , X n ) for j < n .Pr( X j ≤ x , X n ≤ x | R j = 1 , R n = 1) = Pr( X j ≤ x , X n ≤ x , X j > M j − , X n > M n − )Pr( X j > M j − , X n > M n − ) . The numerator can be written asPr( X j ≤ x , X n ≤ x , X j > M j − , X n > M n − )= (cid:90) x −∞ (cid:90) min( x ,z ) −∞ Pr (cid:0) M j − < z , M n − j +1 < z | X j = z , X n = z (cid:1) φ ( z ; ¯ Σ I , I )d z d z = (cid:90) x −∞ (cid:90) x −∞ Pr (cid:0) M j − < z , M n − j +1 < z | X j = z , X n = z (cid:1) φ ( z ; ¯ Σ I , I ) ( z > x )d z d z + (cid:90) x −∞ (cid:90) z −∞ Pr (cid:0) M j − < z , M n − j +1 < z | X j = z , X n = z (cid:1) φ ( z ; ¯ Σ I , I ) ( z < x )d z d z = (cid:90) x x (cid:90) x −∞ Pr (cid:0) M j − < z , M n − j +1 < z | X j = z , X n = z (cid:1) φ ( z ; ¯ Σ I , I )d z d z + (cid:90) x −∞ (cid:90) z −∞ Pr (cid:0) M j − < z , M n − j +1 < z | X j = z , X n = z (cid:1) φ ( z ; ¯ Σ I , I )d z d z = (cid:90) x −∞ (cid:90) x −∞ Φ n − (cid:0) (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) , I (cid:123) ; I (cid:1) φ ( z ; ¯ Σ I , I )d z d z − (cid:90) x −∞ (cid:90) x −∞ Φ n − (cid:0) (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) , I (cid:123) ; I (cid:1) φ ( z ; ¯ Σ I , I )d z d z + (cid:90) x −∞ (cid:90) z −∞ Φ n − (cid:0) (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) , I (cid:123) ; I (cid:1) φ ( z ; ¯ Σ I , I )d z d z where (cid:37) I (cid:123) , I (cid:123) is as in (18). We multiply and divide each term within the integrals withΦ n − ( ; Γ I (cid:123) , I (cid:123) ). Then, we recognize that the first two integrals provide the distributionof the closed skew-normal random vector we introduced before, evaluated at the points( x , x ), ( x , x ). Instead, the third integral represents the distribution of the random vector( Z − Z , Z ) which again according to (4) follows a closed skew-normal distribution, i.e., (cid:32) −
10 1 (cid:33) (cid:32) Z Z (cid:33) = D (cid:32) Z Z (cid:33) ∼ CSN ,n − ( D ¯ Σ I , I D (cid:62) , ∆ , ¯ Σ ∗∗I (cid:123) , I (cid:123) ; I ) , where ∆ = (cid:37) I (cid:123) , I (cid:123) ¯ Σ I (cid:123) , I (cid:123) D (cid:62) ( D ¯ Σ I , I D (cid:62) ) − and ¯ Σ ∗∗I (cid:123) , I (cid:123) ; I = Γ I (cid:123) , I (cid:123) − ∆ D ¯ Σ I , I (cid:37) (cid:62)I (cid:123) , I (cid:123) .18t follows from Theorem 2.9 that the two events: a record occuring at time j and n , arenot independent. Indeed, the probability Φ n − ( ; ˜ Ω ) is different from the product of the twomarginal probabilities Φ j − ( ; Γ j − j − ) and Φ n − ( ; Γ n − n − ), derived in Proposition3.1. Remark . The marginal distribution of X j , given that ( X j , X n ) are records, is Pr( X j ≤ x | R j = 1 , R n = 1)= Φ n − ( ; Γ I (cid:123) , I (cid:123) )Φ n − ( ; ˜ Ω ) × (cid:0) Ψ ,n − ( x ; 1 , ∆ , ¯ Ω I (cid:123) , I (cid:123) ; I ) − Ψ ,n − ( x , x ; ¯ Σ I , I , (cid:37) I (cid:123) , I (cid:123) , ¯ Σ I (cid:123) , I (cid:123) ; I )+Ψ ,n − (0 , x ; D ¯ Σ I , I D (cid:62) , ∆ , ¯ Σ I (cid:123) , I (cid:123) ; I ) (cid:1) . where ∆ = (cid:32) σ − ii (1 − ρ ij ) i =1 ,...,j − σ − ii ( ρ jn − ρ in ) i = j +1 ,...,n − (cid:33) , ¯ Ω I (cid:123) , I (cid:123) ; I = ¯ Σ I (cid:123) , I (cid:123) ; I + (cid:32) σ − ii ( ρ ij ρ jn − ρ in ) i =1 ,...,j − σ − ii (1 − ρ jn − ρ ij − ρ in ρ jn ) i = j +1 ,...,n − (cid:33) , and these parameters are obtained from (4) , with A := (0 1) . See the proof of Theorem 2.9for the details. Hence, similarly to the case of independent random variables in Falk et al.(2018), the distribution of X j being a record is affected, if we know that X n is a record aswell. The marginal distribution of X n , given that ( X j , X n ) are records, is Pr( X n ≤ x | R j = 1 , R n = 1) = Φ n − ( ; Γ I (cid:123) , I (cid:123) )Φ n − ( ; ˜ Ω ) Ψ ,n − (0 , x ; D ¯ Σ I , I D (cid:62) , ∆ , ¯ Σ I (cid:123) , I (cid:123) ; I ) Hence, different to the case of independent random variables in Falk et al. (2018) we havethat the distribution of X n , being a record, is affected by the additional knowledge that attime j < n there was a record. Although stationary Gaussian sequences are useful for a wide range of statistical analysis(e.g., Lindgren 2012, Brockwell and Davis 2013, Banerjee, Carlin, and Gelfand 2014, Cressieand Wikle 2015, to name a few), a natural question that arises is the following. What are theproperties of records for a stationary sequence of dependent rvs when the univariate marginaldistribution, F , is non-Gaussian? This question is even more relevant if it is assumed that19 is unknown, which concerns many real-world applications. Some of the previous resultsare clearly independent of the underlying df F , provided it is continuous. The probabilitythat X n is a record, or the distribution of the arrival time of the n -th record, for example, donot depend on F . The distribution of X n , conditional to the assumption that it is a record,however does depend on F .Let { X n , n ≥ } be a strictly stationary sequence of rvs, i.e. the joint distributionof ( X j , . . . , X j n ) and ( X j + m , . . . , X j n + m ) are identical, for every n, m and j , . . . , j n . Weprovide an answer to the above question under some restrictions on the tail behavior ofthe marginal distribution of such a process and on the dependence structure. Precisely, weassume that F belongs to the (maximum) domain of attraction of G γ , in symbols F ∈ D ( G γ ), γ ∈ R . This means that, if Y , . . . , Y n are iid rv with common cdf F , then there existsequences of norming constants a n > b n ∈ R such thatlim n →∞ Pr(max( Y , . . . , Y n ) ≤ a n x + b n ) = lim n →∞ F n ( a n x + b n ) = G γ ( x ) , x, γ ∈ R . (19)This cdf is the Generalized Extreme-Value (GEV) class of distributions. The cdfs of thethree sub-classes of the GEV, i.e. the Gumbel, Fr´echet, and negative Weibull are denotedby G ( x ), G α ( x ) = G /γ (( x − /γ ) for γ > G β ( x ) = G − /γ ( − ( x + 1) /γ ) for γ < { X n , n ≥ } we assume a mild condition onthe long-range dependence of extremes of such a stationary sequence. Precisely, we assumethat { X n , n ≥ } is a stationary sequence with a univariate marginal df F that satisfies F ∈ D ( G γ ), γ ∈ R and according to Leadbetter et al. (1983, Ch. 3) the following dependencerestriction is required. Partition { , . . . , n } into k n = (cid:98) n/r n (cid:99) blocks of length r n = o ( n ).Suppose that for every λ >
0, there is a sequence of real-value thresholds u n ( λ ), n = 1 , . . . ,such that lim n →∞ n Pr( X > u n ( λ )) = λ, λ > D ( u n ( λ )) is satisfied for each such λ . Specifically, for every λ >
0, let K n ( l ) = max( | Pr( X i ≤ u n ( λ ) , i ∈ I ∪ J ) − Pr( X i ≤ u n ( λ ) , i ∈ I ) Pr( X i ≤ u n ( λ ) , i ∈ J ) | )where I, J ⊂ { , . . . , n } such that min {| i − j | : i ∈ I, j ∈ J } = l . Then, we say that condition D ( u n ( λ )) holds for each such λ , if K n ( l n ) → n → ∞ for some sequence l n → ∞ with l n = o ( n ) (pp. 53-57, Leadbetter et al. 1983). By Lemma 3.2.2 in Leadbetter et al. (1983)20his means that extreme events, such as the partial maxima M E i = max j ∈ E i ( X j ), with E i = { ( i − r n + 1 , . . . , ir n } \ { ir n − l n + 1 , . . . , ir n } , i = 1 , . . . , k n , which are separated by l n , are almost independent.Then, under these conditions by Leadbetter et al. (1983, Theorem 3.7.1) we have that forsuitable choices of r n → ∞ with n → ∞ such that k n K n ( l n ) → k n l n → n → ∞ ,it follows thatlim n →∞ Pr(max( X , . . . , X n ) ≤ u n ( λ )) = exp( − θλ ) , < θ ≤ , λ > . (20)When this holds true we say that the sequence { X n , n ≥ } has extremal index θ ∈ (0 , λ = − log G γ ( x ), x ∈ R , and suitable norming constants a n > b n ∈ R ,lim n →∞ Pr (max( X , . . . , X n ) ≤ a n x + b n ) = G γ ( x ) θ , x, γ ∈ R , Loosely speaking, the extremal index is a parameter that quantifies the impact thatthe dependence structure of the stationary sequence has on the asymptotic distribution ofextreme events such as the partial maximum M n , for sufficiently large n . When θ = 1 werecover (19), i.e. the asymptotic distribution of the normalized maximum for a sequence ofindependent variables. When θ <
1, then for every x ∈ R we have that G γ ( x ) ≤ G θγ ( x ) andtherefore 1 − G γ ( x ) ≥ − G θγ ( x ). In other words, the dependence of the stationary sequencereduces the size of the extreme events. Theorem 2.11.
Let { X n , n ≥ } be a stationary sequence that has extremal index < θ ≤ .Then, n Pr( R n = 1) → θ − , as n → ∞ . (21) Furthermore, there are sequences of norming constants a n > and b n ∈ R such that theasymptotic distribution of X n (suitably normalized), given that it is a record, is lim n →∞ Pr( X n ≤ a n x + b n | R n = 1) = G θγ ( x ) , x, γ ∈ R , < θ ≤ . (22) Proof.
First, we show that there are on average approximately θ − records among X , . . . , X n ,for large n . Precisely, (21) is obtained from n Pr( X n > M n − ) = n (cid:90) supp( F ) Pr( M n − < v | X n = v ) f X n ( v )d u = (cid:90) Pr( M n − < u n ( t ) | X n = u n ( t )) (cid:124) (cid:123)(cid:122) (cid:125) A n × na n f X n ( u n ( t )) ( D ) (cid:124) (cid:123)(cid:122) (cid:125) B n d t v = u n ( t ) with u n ( t ) = a n t + b n and D := { t ∈ R : u n ( t ) ∈ supp( F ) } . By Theorem 3.7.1 in Leadbetter et al. 1983, for any t ∈ D , we havePr( M n − ≤ u n ( t ) , X n ≤ u n ( t )) = Pr( M n − ≤ u n ( t ) | X n ≤ u n ( t ))(1 − Pr( X n > u n ( t )))= Pr( M n − ≤ u n ( t ) | X n ≤ u n ( t ))(1 + o (1)) , and, on the other hand, we havePr( M n − ≤ u n ( t )) ≥ Pr( M n − ≤ u n ( t ) , X n ≤ u n ( t )) ≥ Pr( M n − ≤ u n ( t )) − Pr( M n − ≤ u n ( t ) , X n > u n ( t )) ≥ Pr( M n − ≤ u n ( t )) − Pr( X n > u n ( t )) ≥ Pr( M n − ≤ u n ( t )) + o (1) . From these two results it follows that for any t ∈ D we havePr( M n − ≤ u n ( t ) | X n ≤ u n ( t )) = Pr( M n − ≤ u n ( t )) + o (1) . (23)By (23) and Theorem 3.7.1 in Leadbetter et al. 1983 (cf. O’Brien 1987, Rootz´en 1988) wehave A n ≈ Pr( M n − < u n ( t )) , n → ∞≈ exp {− n Pr( X > u n ( t )) Pr ( M r n ≤ u n ( t ) | X > u n ( t )) } , as n → ∞ . Note that n Pr( X > u n ( t )) n →∞ −→ V ( t ) = t − α , t > , α < , if F ∈ D ( G α ) , ( − t ) β , t < , β > , if F ∈ D ( G β ) ,e − t , t ∈ R , if F ∈ D ( G ) , and Pr ( M r n ≤ u n ( t )) | X > u n ( t )) n →∞ −→ θ. By Resnick (1987, Ch. 2) we have B n n →∞ −→ g ( t ) ( t ∈ supp( G γ )) = αt − ( α +1) , t > , α < , if F ∈ D ( G α ) ,β ( − t ) β − , t < , β > , if F ∈ D ( G β ) ,e − t , t ∈ R , if F ∈ D ( G ) . n → ∞ , n Pr( X n > M n − ) ≈ (cid:90) supp( G γ ) exp {− V ( t ) θ } g ( t )d t = θ − , and hence, (21) is proven.Finally, using similar arguments we obtainPr( X n ≤ a n x + b n | R n = 1) = Pr( X n ≤ a n x + b n , X n > M n − )Pr( X n > M n − )= n (cid:82) v ∈ supp( F ): v ≤ a n x + b n Pr( M n − < v | X n = v ) f X n ( v )d un Pr( X n > M n − ) n →∞ −→ G θγ ( x ) , x, γ ∈ R and the proof is complete.Theorem 2.11 states that for a stationary sequence of dependent rvs { X n , n ≥ } , underappropriate conditions on the dependence structure, the asymptotic distribution of X n (ap-propriately normalized), being a record, coincides with the asymptotic distribution G θγ of thenormalized maximum. This finding generalizes Lemma 2.1 in Falk, Chokami, and Padoan(2018), derived for a sequence of indepedent rvs. Indeed, the same result is obtained for θ = 1.In the following part the are three specific examples of asymptotic distributions of recordsthat stem from the general formula (22) in Theorem 2.11. Example . For an integer m ≥ , let { ε n , n ≥ } be a sequence ofiid rvs uniformly distributed on { , /m, . . . , ( m − /m } . Let X be a rv uniformly distributedon [0 , , being independent of { ε n } . The process X n = m − X n − + ε n , n ≥ , defines a strictly stationary first-order autoregressive sequence. For n = 1 , , . . . take thenorming constants a n > /n and b n = 1 . Then, lim n →∞ Pr( X n ≤ x/n | R n = 1) = e θ x , x < , where θ = ( m − /m with m ≥ . xample . Let { X n,i , n ≥ , i ≥ } be a triangular array of rvs suchthat for every n { X n,i , i ≥ } is a SSG sequence. Define ρ n,j = E( X n,i X n,i + j ) with i ≤ n and j ≥ . Assume that (1 − ρ n,j ) log n → δ j ∈ (0 , ∞ ] for all j ≥ as n → ∞ . For n = 1 , , . . . choose the norming constants a n = (2 log n ) − / and b n = a − n − a n (log log n + log 4 π ) / Then, lim n →∞ Pr( X n ≤ a n x + b n | R n = 1) = e − θe − x , x ∈ R , where θ = E U (cid:26) Φ | K | (cid:18)(cid:112) δ k − U √ δ k ; Σ (cid:19)(cid:27) and where K = { k ∈ A ⊂ { , . . . } : δ k < ∞} , U is a standard exponential rv and Σ is acorrelation matrix with upper diagonal entries δ i + δ j − δ | i − j | (cid:112) δ i δ j , ≤ i < j ≤ | K | . Example . Let ε , ε , . . . be iid stable (1 , α, κ ) rvs.We recall that a rv is stable ( τ, α, κ ) with τ ≤ , < α ≤ and | κ | ≤ if its characteristicfunction is ω ( x ) = exp (cid:26) − τ α | x | α (cid:18) − i κh ( x, α ) x | x | (cid:19)(cid:27) where i = − and h ( x, α ) = tan( πα/ for α (cid:54) = 1 and h ( x,
1) = 2 π − log | x | otherwise. Let { c i , i ∈ Z } be a sequence of constants satisfying (cid:80) ∞ i = −∞ | c i | α < ∞ and (cid:80) ∞ i = −∞ c i log | c i | isconvergent for α = 1 and κ (cid:54) = 0 . Define the moving average process X n = ∞ (cid:88) i = −∞ c i ε n − i , n ≥ . For n = 1 , , . . . choose the norming constants a n = n /α and b n = 0 . Then, lim n →∞ Pr( X n ≤ xn /α | R n = 1) = e − θx − α , x > , where θ = k α ( c α + (1 + κ ) + c α − (1 − κ )) , with c ± = max −∞
1? We know thatPr( X n > M n − ) ≈ nθ , n → ∞ . Therefore, by elementary arguments,E( N ) = ∞ (cid:88) n =1 Pr( X n > M n − ) n →∞ −→ ∞ . Let { X n , n ≥ } be a sequence of d -dimensional random vectrors X n = ( X (1) n , . . . , X ( d ) n ) ∈ R d . We recall that the rv X n is a complete record if X n > max( X , . . . , X n − ) wherethe maximum is computed componentwise. Here we consider a second-order stationarymultivariate Gaussian process and we extend some of the results derived in Section 2.2 tothe multivariate case. Precisely, we study the probability that a complete record X n occursand the distribution of X n (being a record). We also study the probability that two completerecords ( X j , X n ) occur and the joint distribution of ( X j , X n ) (being records). Without lossof generality, assume for simplicity that E( X i ) = 0, E( X i ) = 1 for every 1 ≤ i ≤ n .Let X = ( X , . . . , X n ) be an nd -dimensional random vector and consider the partition X = ( X (cid:62)I , X (cid:62)I (cid:123) ) (cid:62) ∼ N nd ( µ , Σ ) with corresponding partition of the parameters µ and Σ .The formula of the conditional distribution of X I (cid:123) given that X I = x I , for all x I ∈ R |I| ,in (5) is still valid with the obvious changes. Further on we will provide the specific detailswhenever we use such a formula. Proposition 3.1.
Let { X n , n ≥ } be a SSG sequence of random vectors in R d . For every n ≥ , the probability that X n is a record and the distribution of X n , given that it is arecord, are equal to Pr( R CR n = 1) = Φ ( n − d ( ; Γ n − ,n ) , (24)Pr( X n ≤ x | R CR n = 1) = Ψ d, ( n − d (cid:0) x ; ¯ Σ n , (cid:37) n − ,n , ¯ Σ n − , n − n (cid:1) , (25) where B = n − n − . . . n − n − n − . . . n − ... ... ... n − n − . . . n − ∈ R ( n − d,d n − ,n = σ − n − , n − n ( B − Σ (1: d )1: n − ,n ¯ Σ − n ) ∈ R ( n − d,d (26) where Σ (1: d )1: n − ,n is the covariance matrix of ( X (1)1 , X (1)2 , . . . , X (1) n − , . . . , X ( d )1 , X ( d )2 , . . . , X ( d ) n − ) and X n , ¯ Σ n is the variance-covariance matrix of X n and Γ n − ,n = ¯ Σ n − , n − n + (cid:37) n − ,n ¯ Σ n (cid:37) (cid:62) n − ,n Proof.
We start deriving the probability that X n is a record.Pr( X n > M n − ) = Pr( X n,i > M n − ,i , i = 1 , . . . , d )= (cid:90) R d Pr( M n − ,i < z i , i = 1 , . . . , d | X n,i = z i , i = 1 , . . . , d ) φ d ( z )d z Let X ( i )1: n − be the vector of the i -th components of X , . . . , X n − . Then,Pr( M n − ,i < z i , i = 1 , . . . , d | X n,i = z i , i = 1 , . . . , d )= Pr( ∩ di =1 { X k,i < z i , k = 1 , . . . , n − }| X n,i = z i , i = 1 , . . . , d )= Pr( X ( i )1: n − < n − z i , i = 1 , . . . , d | X n = z ) . By the multivariate version of the conditional Gaussian distribution in (5) we have( X , X , . . . , X n − | X n = z n ) ∼ N ( n − d (cid:0) µ n − ,n , Σ n − , n − n (cid:1) where µ n − ,n = (cid:16) Σ ( i )1: n − ,n ¯ Σ − n (cid:17) i =1 ,...,d z ∈ R ( n − d Σ n − , n − n = (cid:16) Σ ( i,h )1: n − , n − − Σ ( i )1: n − ,n ¯ Σ − n Σ ( h ) (cid:62) n − ,n (cid:17) i,h =1 ,...,d . Σ ( i )1: n − ,n is the covariance matrix of X ( i )1: n − and X n , and Σ ( i,h )1: n − , n − is the covariance matrixof X ( i )1: n − and X ( h )1: n − . We have thatPr( X ( i )1: n − < n − z i , i = 1 , . . . , d | X n = z ) = Φ ( n − d ( (cid:37) n − ,n z )where (cid:37) n − ,n is defined as in equation (26). ThereforePr( X n > M n − ) = (cid:90) R d Φ ( n − d ( (cid:37) n − ,n z ) φ d ( z )d z = E (cid:0) Φ ( n − d ( (cid:37) n − ,n Z ) (cid:1) , Z ∼ N ( n − d ( , ¯ Σ n − , n − n ) and the claim follows by applying Proposition 7.1 in(Azzalini and Capitanio 1999).The computation of the distribution function follows the same procedure. We need tocomputePr( X n ≤ x , X n > M n − ) = (cid:90) ( −∞ , x ] Pr( M n − ,i < z i , i = 1 , . . . , d | X n,i = z i , i = 1 , . . . , d ) φ d ( z )d z = Φ ( n − d ( ; ¯ Σ n − n + (cid:37) n − ,n ¯ Σ n (cid:37) (cid:62) n − ,n ) × Ψ d, ( n − d ( x ; ¯ Σ n , (cid:37) n − ,n , ¯ Σ n − n ) Remark . For every n = 1 , . . . , m < n and ≤ i < j ≤ d , if Cor( X ( i ) n , X ( j ) n ) = 0 and Cor( X ( i ) n , X ( i ) m ) = 0 , then we obtain Pr ( X n > M n − ) = Φ ( n − d ( ; ¯ Σ n − , n − n + (cid:37) n − ,n (cid:37) (cid:62) n − ,n ) where the variance-covariance matrix is a diagonal block matrix, with each diagonal blockbeing equal to I n − + n − (cid:62) n − . Since we have d diagonal blocks, we obtain Φ ( n − d ( ; ¯ Σ n − , n − n + (cid:37) n − ,n (cid:37) (cid:62) n − ,n ) = (cid:0) Φ n − ( ; I n − + n − (cid:62) n − ) (cid:1) d = n − d by the result in Remark 2.2. Our next result deals with the joint distribution of two complete records at times j and n > j . In the following, we use the notation x J , J ⊆ { , . . . , d } to indicate a vector ofdimension |J | whose components are the entries of x ∈ R d determined by the elements of J . Theorem 3.3.
Let { X n , n ≥ } be a SSG sequence of random vectors in R d . For j and n > j , set I = { j, n } . Then Pr( R CR j = 1 , R CR n = 1) = Φ d ( n − ( ; Γ I (cid:123) I (cid:123) )Ψ d,d ( n − ( ; L ) (27)Pr( X j ≤ x , X n ≤ x | R CR j = 1 , R CR n = 1)= (cid:80) J ⊆{ ,...,d } Ψ d,d ( n − ( J , x J , x J , x J ; L J ) − Ψ d,d ( n − ( J , x J , x J , x J ; L J )Ψ d,d ( n − ( ; L ) , (28) where Ψ m,q ( · ; L ) ∼ CSN m,q ( L ) , Γ I (cid:123) I (cid:123) := ¯ Σ I (cid:123) I (cid:123) ; I + (cid:37) n − ,n ¯ Σ II (cid:37) (cid:62) n − ,n , L ( · ) = (cid:0) D ( · ) ¯ Σ II D (cid:62) ( · ) , ∆ ( · ) , ¯ Σ I (cid:123) I (cid:123) ; I (cid:1) , (29)27 ( · ) = (cid:37) n − ,n ¯ Σ I (cid:123) , I (cid:123) D (cid:62) ( · ) ( D ( · ) ¯ Σ I , I D (cid:62) ( · ) ) − and D = (cid:16) I d − I d (cid:17) D J = I J ¯ J − I J ¯ J J I ¯ J J ¯ J J ¯ J I J ¯ J J ¯ J J I ¯ J Proof.
We computePr( R CR j = 1 , R CR n = 1)= Pr( X j > M j − , X n > M n − )= Pr( X j > M j − , X n > M j +1: n − , X j < X n )= (cid:90) R d (cid:90) ( − ∞ , z n ] Pr( M j − < z j , M j +1: n − < z n | X i = z i , i ∈ I ) φ d ( z j , z n ; ¯ Σ II )d z j d z n First of all, we recall the inverse blok-matrix of a two-by-two block matrix:¯ Σ − II = (cid:32) Λ − ¯ Σ − j Σ j,n Λ − Λ Σ n,j ¯ Σ − j Λ (cid:33) where Λ i is the Schur complement of ¯ Σ i in ¯ Σ II , for i ∈ I , for example, Λ = (cid:16) ¯ Σ j − Σ j,n ¯ Σ − n Σ n,j (cid:17) − .By the multivariate version of the conditional Gaussian distribution in (5) we have that( X i , i ∈ I (cid:123) ) | ( X i = z i , i ∈ I ) ∼ N | I (cid:123) | (cid:0) µ I (cid:123) I (cid:123) ; I , Σ I (cid:123) I (cid:123) ; I (cid:1) . Specifically we have µ I (cid:123) I (cid:123) ; I = (cid:0) ( µ (cid:62) i , i = 1 , . . . , d ) , ( µ (cid:62) h , h = d + 1 , . . . , d ) (cid:1) (cid:62) with µ i = (cid:16) Σ ( i )1: j − ,j Λ − Σ ( i )1: j − ,n Λ Σ n,j ¯ Σ − j (cid:17) z j + (cid:16) − Σ ( i )1: j − ,j ¯ Σ − j Σ j,n Λ + Σ ( i )1: j − ,n Λ (cid:17) z n =: µ ij z j + µ in z n , (30) µ h = (cid:16) Σ ( h − d ) j +1: n − ,j Λ − Σ ( h − d ) j +1: n − ,n Λ Σ n,j ¯ Σ − j (cid:17) z j + (cid:16) − Σ ( h − d ) j +1: n − ,j ¯ Σ − j Σ j,n Λ + Σ ( h − d ) j +1: n − ,n Λ (cid:17) z n =: µ hj z j + µ hn z n , (31)and Σ I (cid:123) I (cid:123) ; I is a (2 d ) × (2 d ) matrix. It is defined by blocks ( Σ I (cid:123) I (cid:123) ; I ) i,h =1 ,..., d of the form( Σ I (cid:123) I (cid:123) ; I ) ih = Σ ( i,h )1: j − , j − − (cid:16)(cid:16) Σ ( i )1: j − ,j Λ − Σ ( i )1: j − ,n Λ Σ n,j ¯ Σ − j (cid:17) Σ ( h )1: j − ,j + (cid:16) − Σ ( i )1: j − ,j ¯ Σ − j Σ j,n Λ + Σ ( i )1: j − ,n Λ (cid:17) Σ ( h )1: j − ,n (cid:17) ih i, h = 1 , . . . , d Σ I (cid:123) I (cid:123) ; I ) ih = Σ ( i,h − d )1: j − ,j +1: n − − (cid:16)(cid:16) Σ ( i )1: j − ,j Λ − Σ ( i )1: j − ,n Λ Σ n,j ¯ Σ − j (cid:17) Σ ( h − d ) j +1: n − ,j + (cid:16) − Σ ( i )1: j − ,j ¯ Σ − j Σ j,n Λ + Σ ( i )1: j − ,n Λ (cid:17) Σ ( h − d ) j +1: n − ,n (cid:17) ih i = 1 , . . . , d, h = d + 1 , . . . , d ( Σ I (cid:123) I (cid:123) ; I ) ih = Σ ( i − d,h ) j +1: n − , j − − (cid:16)(cid:16) Σ ( i − d ) j +1: n − ,j Λ − Σ ( i − d ) j +1: n − ,n Λ Σ n,j ¯ Σ − j (cid:17) Σ ( h )1: j − ,j + (cid:16) − Σ ( i − d ) j +1: n − ,j ¯ Σ − j Σ j,n Λ + Σ ( i − d ) j +1: n − ,j Λ (cid:17) Σ ( h )1: j − ,n (cid:17) ih i = d + 1 , . . . , d, h = 1 , . . . , d ( Σ I (cid:123) I (cid:123) ; I ) ih = Σ ( i − d,h − d ) j +1: n − ,j +1: n − − (cid:16)(cid:16) Σ ( i − d ) j +1: n − ,j Λ − Σ ( i − d ) j +1: n − ,n Λ Σ n,j ¯ Σ − j (cid:17) Σ ( h − d ) j +1: n − ,j + (cid:16) − Σ ( i − d ) j +1: n − ,j ¯ Σ − j Σ j,n Λ + Σ ( i − d ) j +1: n − ,j Λ (cid:17) Σ ( h − d ) j +1: n − ,n (cid:17) ih i, h = d + 1 , . . . , d Therefore, we obtainPr( X ( i )1: j − < j − z j , X ( i ) j +1: n − < n − j − z n , i = 1 , . . . , d | X i = z i , i ∈ I ) = Pr( Z < (cid:37) I (cid:123) , I (cid:123) z )where Z ∼ N ( n − d ( , ¯ Σ I (cid:123) I (cid:123) ; I ), (cid:37) I (cid:123) , I (cid:123) = σ − I (cid:123) , I (cid:123) ; I ( B − Σ (1: d ) I (cid:123) , I ¯ Σ − II ) ∈ R ( n − d,d (32) B = j − j − . . . j − j − j − . . . j − j − j − . . . j − j − j − . . . j − ... ... ... ... ... ... j − j − . . . j − j − j − . . . j − n − j − n − j − . . . n − j − n − j − n − j − . . . n − j − n − j − n − j − . . . n − j − n − j − n − j − . . . n − j − ... ... ... ... ... ... n − j − n − j − . . . n − j − n − j − n − j − . . . n − j − and z = ( z j , z n ) is a column vector with length 2 d . We obtainPr( R CR j = 1 , R CR n = 1)= (cid:90) R d (cid:90) ( − ∞ , z n ] Φ d ( n − ( (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) I (cid:123) ; I ) φ d ( z j , z n ; ¯ Σ II )d z j d z n = Φ d ( n − ( ; ¯ Σ I (cid:123) I (cid:123) ; I + (cid:37) I (cid:123) , I (cid:123) ¯ Σ II (cid:37) (cid:62)I (cid:123) , I (cid:123) ) Pr( Z < Z )29here ( Z , Z ) ∼ CSN d,d ( n − ( ¯ Σ II , (cid:37) I (cid:123) , I (cid:123) , ¯ Σ I (cid:123) I (cid:123) ; I ). Formula (27) follows by noting that Z − Z = (cid:16) I d − I d (cid:17) (cid:32) Z Z (cid:33) and by applying (4)To compute formula (28), we repeat the same procedure. With D ( a , . . . , a n ) := (cid:81) ( −∞ , a i ]we obtainPr( X j ≤ x , X n ≤ x , R CR j = 1 , R CR n = 1)= Pr( X j ≤ x , X n ≤ x , X j > M j − , X n > M j +1: n − X i , X j < X n )= (cid:90) ( − ∞ , x ] (cid:88) J ⊆{ ,...,d } (cid:32)(cid:90) D ( z n J , x J ) Pr( M j − < z j , M j +1: n − < z n | X i = z i , i ∈ I ) φ d ( z j , z n ; ¯ Σ II )d z j (cid:33) ( z n J < x J , z n ¯ J > x J )d z n = (cid:88) J ⊆{ ,...,d } (cid:90) ( − ∞ , x J ] (cid:90) ( x J , x J ] (cid:90) D ( z n J , x J ) Φ d ( n − ( (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) I (cid:123) ; I ) φ d ( z j , z n ; ¯ Σ II )d z j d z n = (cid:88) J ⊆{ ,...,d } (cid:32)(cid:90) D ( x J , x J , z n J , x J ) Φ d ( n − ( (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) I (cid:123) ; I ) φ d ( z j , z n ; ¯ Σ II )d z j d z n − (cid:90) D ( x J , x J , z n J , x J ) Φ d ( n − ( (cid:37) I (cid:123) , I (cid:123) z ; ¯ Σ I (cid:123) I (cid:123) ; I ) φ d ( z j , z n ; ¯ Σ II )d z j d z n (cid:33) = (cid:88) J ⊆{ ,...,d } Φ d ( n − ( ; ¯ Σ I (cid:123) I (cid:123) ; I + (cid:37) I (cid:123) , I (cid:123) ¯ Σ II (cid:37) (cid:62)I (cid:123) , I (cid:123) )(Pr( Z J < Z J , Z J ≤ x J , Z J ≤ x J , Z J ≤ x J ) − Pr( Z J < Z J , Z J ≤ x J , Z J ≤ x J , Z J ≤ x J ))The first probability on the right-hand side can be computed by noting that Z J − Z J Z J Z J Z J = I J ¯ J − I J ¯ J J I ¯ J J ¯ J J ¯ J I J ¯ J J ¯ J J I ¯ J Z J Z J Z J Z J where ( Z , Z ) = ( Z J , Z J , Z J , Z J ) ∼ CSN d,d ( n − ( ¯ Σ II , (cid:37) I (cid:123) , I (cid:123) , ¯ Σ I (cid:123) I (cid:123) ; I ) and by apply-ing (4). 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