Certain bivariate distributions and random processes connected with maxima and minima

It is well-known that [S(x)]n and [F (x)]n are the survival function and the distribution function of the minimum and the maximum of n independent, identically distributed random variables, where S and F are their common survival and distribution functions, respectively. These two extreme order statistics play important role in countless applications, and are the central and well-studied objects of extreme value theory. In this work we provide stochastic representations for the quantities [S(x)]α and [F (x)]α, where α > 0 is no longer an integer, and construct a bivariate model with these margins. Our constructions and representations involve maxima and minima with a random number of terms. We also discuss generalizations to random process and further extensions. AMS 2000 subject classifications:


Introduction
Let {X i } be a sequence of independent and identically distributed (IID) random variables with cumulative distribution function (CDF) which are crucial in countless applications, are the central and well-studied objects of extreme value theory (see, e..g., monographs [2,3,8,9,11,16,19,26,27] and extensive references therein as well as in [15]). It is well-known (see, e.g, [1] or [6]) that the SF of S n and the CDF of Nevertheless, many interesting, flexible, and useful univariate families of distributions have been defined this way over the years, with earliest works connected with an exponential model going back to the first half of the nineteenth century (see, e.g., [10]). In addition to the exponential, these new classes of distributions -often referred to as exponentiated distributions -include those connected with gamma, Pareto, and Weibull laws (see, e.g., [12,13,14,21,22,24] and references therein).
In this work we study a bivariate generalization of the joint distribution (1.3) in the same spirit, where the margins are given by (1.4) and the bivariate model reduces to (1.3) when α is an integer. Moreover, the bivariate model with a non-integer α, as well as the margins (1.4), are constructed through maxima and minima. This generalization does not arise by simply replacing the integer n by α in the bivariate CDF (1.3), as the latter fails to be a CDF for α ∈ (0, 1). Instead, the model follows a construction involving random maxima and minima of IID random variables, preserving the spirit of the vector (S n , M n ).
Our work begins with Section 2, which introduces new stochastic representations for random variables given by (1.4) through random maxima and minima. These leads to new 3 representations of the variables with exponentiated distributions based on exponential, gamma, Weibull, Pareto and other models. These results are instrumental in defining a bivariate model presented in Section 3. Generalizations to random process and further extensions are presented in Section 5. The last section contains proofs and auxiliary results.

Univariate distributions
As before, let {X i } be a sequence of IID random variables with CDF F and SF S.
Further, let N be a non-negative, integer-valued random variable with PDF p(n) = P(N = n) and probability generating function (PGF) G N (s) = Es N , |s| ≤ 1. The following lemma is crucial in obtaining representations of random variables with the SF or the CDF given in (1.4) through random maxima and minima. Here, and throughout the paper, we shall use the convention that the minimum and the maximum over an empty set are ∞ and −∞, respectively.
Lemma 2.1. In the above setting, if N is independent of the sequence {X i }, then the CDF and the SF of the random variables are given by F X (x) = G N (F (x)) and S Y (y) = G N (S(y)), respectively.
We shall use this result to obtain representations of random variables with the SF and the CDF given in (1.4) through random maxima and minima with suitably chosen N .
Remark 2.2. Expressing the function (2.2) as a power series, makes it clear that this is not a PGF when α > 1, as then not all coefficients of s n in the above series are non-negative. For α ∈ (0, 1] this variable represents the number of trials till the first success in an infinite sequence of independent Bernoulli trials where the nth trial is a success with probability α/n. Clearly, at the boundary value α = 1, the distribution is concentrated at n = 1. Let us also mention that in the special case α = 1/2, we obtain the distribution of Z +1, where Z has a discrete Mittag-Leffler distribution with the PGF G(s) = [1 + c(1 − s) α ] −1 with c = 1 (see, e.g., [25]).
In view of this discussion, we obtain the following result.
where N α has the Sibuya distribution (2.3) and is independent of the IID {X j } with the CDF F .
Let us note that that in view of (2.5) the variables X and Y from Corollary 2.3 satisfy the relation X ≥ Y , which is consistent with the following lemma. An extension of Corollary 2.3 to the case α ∈ (0, ∞) is rather straightforward. In the following result and throughout the paper we shall use the notation (2.6) {t} = k and t = t − k whenever k < t ≤ k + 1, k = 0, 1, 2, . . . .
where N t has the Sibuya distribution (2.3) with parameter α = t and is independent of the IID {X j } with the CDF F .

Bivariate models
An extension to bivariate distribution having margins with SF and CDF given by (1.4) is straightforward. One such distribution is obtained by putting together the X and Y defined in Corollary 2.3. Let us start with the following result, which extends Lemma 2.1 to the bivariate setting.
where N is an integer-valued random variable N supported on N = {1, 2, . . .} and the {X i } are IID random variables, independent of N . Then, the CDF of (X, Y ) is given by where G N (·) and F (·) are the PGF of N and the common CDF of the {X i }, respectively. 6 According to Lemma 2.1, the marginal CDF of X in where N α is Sibuya distributed given by the PGF (2.2) with α ∈ (0, 1] and the {X i } are IID random variables, independent of N α . Then, the joint CDF of (X, Y ) is given by where S(·) and F (·) are the common SF and CDF of the {X i }, respectively. Remark 3.3. In the special case α = 1, the joint CDF (3.4) becomes and it describes the random vector (X, X), to which (3.3) reduces in this case (since N α = 1 when α = 1). Note that this CDF is also a special case n = 1 of the joint CDF of the two extreme order statistics, S n and M n , given by (1.3).
Observe that replacing the n in the joint CDF (1.3) of the two extreme order statistics with α ∈ (0, 1) does not produce the CDF (3.4). Moreover, the resulting quantity is not even a proper CDF, as can be easily verified. Our next result provides a generalization of the bivariate CDF (1.3) to non-integer values of n.
and define where N α has the Sibuya distribution (2.3), and is independent of the IID {X j } with the CDF F . Then, the joint CDF of (X, Y ) is where I A (·) is the indicator function of the set A.
where S(·) and F (·) are the common SF and CDF of the respectively. This can also be seen by taking the limits x → ∞ or y → ∞ in (3.7).
Moreover, when t ∈ N, the CDF (3.7) coincides with (1.3). Additionally, when n = 0, the indicator function in (3.7) takes on the value of 1, and we obtain (3.4). Formally, the CDF (3.7) is well defined for any non-negative integer n and any α ∈ [0, 1]. For the boundary cases α = 0 or α = 1, we recover the bivariate CDF of the extreme order statistics.
If the bivariate distributions discussed above are continuous, they can also be described through the copula representation (see [31]) Straightforward calculations show that the copula C N connected with the bivariate distribution (3.2) is given by In the Sibuya case, the copula C α = C Nα connected with the bivariate distribution (3.3) is given by Similarly, in the more general case of (3.6) with n ≥ 1, the copula is This is a generalization of the min-max copula discussed in [28], to which it reduces for boundary values α = 0, 1.

Further examples and a duality of distributions on N
Numerous new families of distributions can be generated from a "base" CDF F (·) (or SF S(·)) via Lemma 2.1 with suitably chosen distribution of N . One well-known example of such a construction is the so-called Marshall-Olkin scheme introduced in [20], where a SF S generates a family of SFs given by For α = p ∈ (0, 1), the above SF is of the form G N (S(x)), where G N is the PGF of a geometric random variable N with parameter p, given by the PDF A straightforward calculation shows that, when α > 1, then the CDF corresponding to and G N is the PGF of a geometric random variable with parameter p = 1/α. Thus, in view of Lemma 2.1, the family of distributions defined viz. (4.1) is generated by geometric minima (0 < α < 1) and geometric maxima (α > 1) of IID random variables with the base SF S, as noted in [20].
Another example of this construction is connected with the so-called transmuted distributions, which have populated the literature since the introduction of the quadratic rank transmutation map in [29]. The latter, defined via is used to transform a base CDF F into its transmuted version F α , where As recently shown in [17], the CDF in (4.4) is of the form G −α (F (x)) for α ∈ [−1, 0] and is the PGF of a Bernoulli distribution with parameter p ∈ [0, 1] shifted up by one. Again, by Lemma 2.1, the family of distributions defined viz. (4.4) is generated by random minima (0 ≤ α ≤ 1) and random maxima (−1 ≤ α ≤ 0) of IID random variables with the base CDF F . See [17] for more information and extensive references.
The above examples illustrate the fact that virtually any integer-valued random variable N generates a new probability distribution from that of a base random variable X In view of of our results in Section 2, it is not hard to see that in order for these conditions to hold the PGFs of N andÑ need to satisfy the relation In turn, if the quantity on the right-hand-side of (4.7) is a genuine PGF, then the process of taking random maxima (or minima) can be "reversed" as described above. This motivates the following definition. Note that the notion of being a dual is symmetric, that is ifÑ is a dual to N then in turn N is a dual ofÑ . Further, as noted above, the four operations of taking random min/max with either N orÑ are reversible and recover the original distribution, as stated in the following result.
with the {X ij } being independent of all the integer valued random variables that appear in , s ∈ [0, 1], 11 which, upon expending the square root above into a power series, produces the dual PDF (4.11) This can be seen as an exponentially tilted Sibuya distribution of N α given by (2.3) with α = 1/2, where r = 4p/(1 + p) 2 .

Negative binomial/negative binomial pair. Consider a negative binomial (NB)
distribution with parameters r ∈ R + and p ∈ (0, 1], given by the PDF After routine calculation, we find the dual PGF to be which we recognize to be a NB distribution with parameters 1/r and p r , truncated below at 1. Thus, NB distributions with parameters r, p and 1/r, p r , both truncated below at 1, form a dual pair in our scheme. In particular, when r = 1, both distributions of the dual pair are the same geometric distributions, given by the PDF (4.2). This has an interesting interpretation: If N is geometrically distributed and Y is obtained via a maximum (minimum) of N IID variables {X i } and in turn we take the minimum (maximum) of N IID copies of Y , then we recover the distribution of the {X i }. respectively, of the ith record among the {nU n }. By assumption, the first value is a record, so that K 1 = 1 and R 1 = U 1 . Since U 1 is less than one with probability one, all the {R i } are less than one as well. Further, we let δ i = R i−1 − R i (with R 0 = 1) represent the differences between successive record values while τ i = K i − K i−1 are the inter-arrival times between successive records. With this notation, the Sibuya process (5.1) admits the

Extensions to random processes
where I A (·) is the indicator function of the set A. Viewing from right to left, the random process N (t) starts with N (1) = 1 and then jumps up by τ i+1 at every record value R i . Other than that, the values of N (t) are constant on the intervals [R n , R n−1 ), and N (R n ) = K n . Twenty samples of the Sibuya process is presented in Figure 1. More information about this process can be found in [18].
The above Sibuya process can be used to construct a bivariate extremal process on where N (t) is the Sibuya process, independent of the {X n }. It follows that for each t ∈ (0, 1) the marginal distributions of (X(t), Y (t)) are given by (3.4) with α = t. In Below we extend this construction beyond the unit interval using Proposition 3.4.

5.1.
A random record process. Consider a sequence X = {X n } of IID, non-negative random variables, and the corresponding upper and the lower records based on these variables, denoted by (K U i (X), R U i (X)) and (K L i (X), R L i (X)), respectively. Thus, K U i is the time (index) at which the ith record occurs among the {X i }, and R U i = X K U i is the size of that record, where a value that is larger than all the previous values sets a new record. The quantities K L i and R L i = X K L i are defined similarly, when a new record is the value that is smaller than the ones previously observed.
We propose a continuous time process {(X(t), Y (t)), t > 0}, containing full information about the records connected with the {X n }, and having certain invariance properties that in our opinion justify naming it the record process. Embedding the records in the record process is similar to embedding the minima S n or the maxima M n (given by (1.1) and (1.2), respectively), into the extremal processes, as described in [27]. However, despite the 14 fact the record process and the extremal processes have the same marginal distributions, they are essentially different, as it will be shown below.
The record process connected with the sequence X = {X n } is defined through a sequence {N (i) (t)}, i ∈ N, of independent Sibuya processes, which are independent of X. Here, each {N (i) (t)} is connected with an underlying sequences U i,n (n ∈ N) of IID standard uniform random variables, as discussed above. As in Section 2, a real t is split into its integer part {t} and its fractional part t = t − {t}, defined in (2.6). Subsequently, we set For t ∈ (0, 1), where we have {t} = 0 and t = t, the minimum 0 j=1 X j and the maximum {t} j=1 X j above are understood as ∞ and −∞, respectively, and (X(t), Y (t)) has the same structure as (5.3). The component X(t) of this construction connects with the lower records (K L i (X), R L i (X)) of the sequence X = {X n }, while the component Y (t) relates to its upper records (K U i (X), R U i (X)). Moreover, by Proposition 3.4, the bivariate marginal distributions of (X(t), Y (t)) are given by (3.7) with n = {t} and α = t . In particular, the the marginal SF of X(t) is [S(·)] t , while the marginal CDF of Y (t) is [F (·)] t for each t ∈ R + . The following result summarizes fundamental properties of the record process (5.4).
Theorem 5.1. Let (X(t), Y (t)) be a random record process (5.4) connected with a sequence X = {X n } of IID, non-negative random variables with the CDF F and the SF S. Then we have the following: (i) For t ∈ R + , the SF of X(t) is given by S t (·) while the CDF of Y (t) is given by F t (·).
(iii) Both, X(t) and Y (t) are a pure jump processes, with right-hand-side continuous trajectories. Moreover, the trajectories of X(t) are non-increasing, while the trajectories of Y (t) are non-decreasing.
(iv) If t = n ∈ N, then X(n) = n k=1 X k and Y (n) = n k=1 X k with probability one.
(v) If t n ∈ (n, n + 1) is the location of the last jump of X on (n, n + 1), then X on (n, t n ) is independent of X(t) on R + \ (n, t n ). Similar property holds for the Y (t).
Remark 5.2. Let us note that, in contrast with the extremal processes described in [27], here X(t) and Y (t) are directly related to the maxima and the minima of the underlying sequence {X n }. Moreover, if the value of X n (n > 1) does not set a new record, so that S n−1 ≤ X n ≤ M n−1 , the trajectories of X(t) and Y (t) are constant on the interval  While a complete analysis of the bivariate process (5.4) is beyond the scope of this paper, below we provide few additional remarks shedding more light on its basic properties. First, let us contrast the properties of the introduced process with some misleading intuition following from extreme value theory. Let us start with a discussion of the structure of the Sibuya process N (t), discussed fully in [18]. Namely, the arrival times (R 1 , . . . , R N ) of the first N jumps can be represented as x > 0. However despite the fact that N (t) is very large when t is close to zero, the process j=1 X j does not lose the effect of the entire SF of X on its SF, which is [S(x)] t . In our special case, this is still Pareto distribution, but with the tail parameter αt, and thus with much heavier power tail than that of the Fréchet distribution.
The above considerations lead to a question about the asymptotics of X(t) at zero. Namely, are there any normalizing functions a(t) and b(t) such has a non-degenerate limiting distribution as t approaches zero? This is normally the case when the random index N (t) tends to infinity while a scaled version of it, c(t)N (t), has a non-zero limiting distribution (see, e.g., [8], Section 4.3). However, as shown below, the Sibuya distribution N (t) increases so fast as t gets close to zero, that no normalization c(t) can make it converge to a non-zero limit. In view of the above result, one may suspect that the quantity Z(t) in (5.5) can never converge to an extreme value (or other non-degenerate) distribution as t approaches zero.
As shown below, this is indeed the case. To see this, consider again standard Pareto distributed {X i } and assume to the contrary that there exist functions a(t) and b(t) such that the distribution of Z(t) has a non-degenerate limit at zero. If this was so, then a(t) would have to converge to infinity, as otherwise b(t) would have to converge to infinity to ensure that the SF 1∧(xa(t)+b(t)) −αt would converge to a SF. However such a limit would be independent of x, and thus it would not constitute a proper distribution. Further, note that b(t)/a(t) must converge to infinity as well, since otherwise 1 ∧ (xa(t) + b(t)) −αt = j=1 X j as t converges to zero.

Proofs
Proof of Lemma 2.1. This result follows from a standard conditioning argument. Indeed, leads to S Y (y) = G N (S(y)), and completes the proof.
This concludes the proof.
Proof of Lemma 3.2. Write F (x, y) = P(X ≤ x) − P(X ≤ x, Y > y), which, by Lemma 2.1, leads to F (x, y) = G N (F (x)) − P(X ≤ x, Y > y). Next, we consider the term P(X ≤ x, Y > y). Clearly, this reduces to zero whenever x ≤ y (since, by the definition of 19 these variables, we have X ≥ Y ). In turn, for x > y, by conditioning on N we obtain X j > y|N = n)P(N = n).
Since the {X i } and N are independent, we have noting that (X, Y ) = (X ∧X α ,Ỹ ∨Y α ) and the two random vectors above are independent.
To complete the proof, substitute the CDFs of (X,Ỹ ) and (X α , Y α ), which are given by X {t}+i . It is clear that Y 1 (t) and Y 2 (t) are independent for each fixed t so that by Corollary 2.3 Both the processes Y 1 and Y 2 are non-decreasing and have right hand side continuous trajectories.
Proof of Proposition 5.3. Suppose that for some deterministic c p > 0 we have convergence in distribution to some (necessarily non-negative) random variable Z stated in the proposition. Expressing this in terms of Laplace transforms (LTs) we will have E(e −tcpNp ) = G p (e −tcp ) → ψ(t) as p → 0 for each t > 0, where ψ(·) is the LT of Z and G p (s) is the PGF of N p given by the righthand-side of (2.2) with α = p. Consequently, we conclude that, for each t > 0, we would In view of this, along with (6.3)-(6.4), we conclude that, for each t > 0, we must have the convergence c p p → 1 − ψ(t) as p → 0, which can not hold unless the right-hand-side is independent on t, which is the case only if P(Z = 0) = 1. This concludes the proof.