2. CHARACTERIZATIONS OF PROCESSES WITH THE UOSP BY MEANS OF [ell ]_∞-SPHERICAL DENSITIES

We start this section with an interesting property of random vectors with [ell ]_∞-spherical densities. It will be seen later that this property is closely related to the UOSP of the underlying random variables.

Proposition 2.1: If X₁,X₂,…,X_n have an [ell ]_∞-spherical density f of the form (1.1), then any k-dimensional marginal of f is also an [ell ]_∞-spherical density, k ≤ n.

Proof: By the exchangeability of f and backward induction, it suffices to prove that the marginal density f_n−1 of (X₁,X₂,…,X_n−1) is [ell ]_∞-spherical. Now, we write

that is, for

, we have that f_n−1 is a function of max_{1≤i≤n−1} x_i only. █

In a similar manner, the following result can be obtained.

Proposition 2.2: If X₁ ≤ X₂ ≤ ··· ≤ X_n have an [ell ]_∞^≤-spherical density as in (1.2), then so do X₁ ≤ X₂ ≤ ··· ≤ X_k for any k ≤ n.

Proposition 2.2 leads to the following natural question of “extendibility”: If X₁ ≤ X₂ ≤ ··· ≤ X_n have an [ell ]_∞^≤-spherical density as in (1.2), can they be the first n variables of a collection X₁ ≤ X₂ ≤ ··· ≤ X_n ≤ X_n+1 that have an [ell ]_∞^≤-spherical density in

? In order to answer this question, denote the joint density of X₁ ≤ X₂ ≤ ··· ≤ X_n as

and supposing that X₁ ≤ X₂ ≤ ··· ≤ X_n ≤ X_n+1 also have an [ell ]_∞^≤-spherical density, denote it by

Obviously,

from which

It follows that X₁ ≤ X₂ ≤ ··· ≤ X_n is [ell ]_∞^≤-spherically extendable if and only if ψ_n is differentiable and nonincreasing.

Regarding the function ψ_n, it is worthwhile to note that it is related to the density function f_Xn of X_n; it is easy to verify that

Next, we show that any random vector with an [ell ]_∞^≤-spherical density has the UOSP, and vice versa.

Proposition 2.3: The random variables X₁ ≤ X₂ ≤ ··· ≤ X_n have an [ell ]_∞^≤-spherical density as in (1.2) if and only if they have the UOSP.

Proof: First, suppose that X₁ ≤ X₂ ≤ ··· ≤ X_n have an [ell ]_∞^≤-spherical density as in (1.2). Fix a t > 0. By Proposition 2.2, it is seen that it suffices to show that the conditional distribution of X₁,X₂,…,X_n−1, given that X_n−1 ≤ t and X_n > t, is the joint distribution of n − 1 order statistics from a uniform(0,t) distribution.

Denote by f_{X1,X2,…,Xn−1|Xn−1≤t,Xn>t} the conditional density of X₁,X₂,…,X_n−1, given that X_n−1 ≤ t and X_n > t. Let 0 ≤ x₁ ≤ x₂ ≤ ··· ≤ x_n−1 ≤ t. Then,

that is, for each t, it is seen that f_{X1,X2,…,Xn−1|Xn−1≤t,Xn>t}(x₁,x₂,…,x_n−1) is constant over the region {(x₁,x₂,…,x_n−1)|0 ≤ x₁ ≤ x₂ ≤ ··· ≤ x_n−1 ≤ t}. In other words, the conditional density f_{X1,X2,…,Xn−1|Xn−1≤t,Xn>t} is the density of n − 1 order statistics from a uniform(0,t) distribution.

Conversely, suppose that X₁ ≤ X₂ ≤ ··· ≤ X_n have the UOSP. Fix a t > 0. Let 0 ≤ x₁ ≤ x₂ ≤ ··· ≤ x_n−1 ≤ t. Then, on one hand,

and, on the other hand,

that is,

Differentiating (2.2) with respect to t, it is seen that when 0 ≤ x₁ ≤ x₂ ≤ ··· ≤ x_n−1 ≤ t, then f (x₁,x₂,…,x_n−1,t) does not depend on the actual values of x₁,x₂,…,x_n−1. █

A straightforward modification of the proof of Proposition 2.3 yields the following result.

Proposition 2.4: The random variables X₁ ≤ X₂ ≤ ··· ≤ X_n have an [ell ]_∞^≤-spherical density as in (1.2) if and only if the conditional distribution of X₁,X₂,…,X_k, given that X_k+1 = t, is the joint distribution of k order statistics from a uniform(0,t) distribution, for all k ≤ n − 1 and t > 0 for which the above conditional distribution is defined.

From Proposition 2.3 we get the following characterization of processes that have the UOSP property.

Theorem 2.5: A point process

has the UOSP if and only if its first n epoch times T₁ ≤ T₂ ≤ ··· ≤ T_n have a joint [ell ]_∞^≤-spherical density for every n ≥ 1 such that P{T_n < ∞} = 1.

We thus see that for a point process N = {N(t), t ∈ [0,∞)}, the following properties are equivalent:

The equivalence of properties (i) and (ii) has already been known in the literature for a long time (see, for instance, references in Puri [15]). It is worthwhile to mention that in a recent proof of the equivalence of properties (i) and (ii) in Hayakawa [8], the equivalence of the first two conditions with property (iii) is essentially implicit. Namely, in the proof of her Theorem 3.2, Hayakawa [8] obtained the fact that the joint density function of the first n interepoch intervals of N, evaluated at the point (y₁,y₂,…,y_n) ≥ 0, depends only on

; in other words, the density is Schur-constant; see (1.3). As we have mentioned earlier, a sequence of interepoch intervals satisfies the above property if and only if the corresponding first n epoch times have a joint [ell ]_∞^≤-spherical density, n ≥ 1.

A sequence of positive random variables {Y₁,Y₂,…,} is called [ell ]₁-isotropic (see, for example, Berman [2] or Hayakawa [8]) if and only if for every n ≥ 1, the random variables

are distributed as a set U₁,U₂,…,U_n of independent uniform(0,1) random variables, conditioned on

. From a result of Berman [2] (given as Theorem 2.1 in Hayakawa [8]), it follows that a sequence of positive random variables {Y₁,Y₂,…} is [ell ]₁-isotropic if and only if the joint density function of any finite number of the Y_i's is Schur-constant. We can therefore restate Theorem 2.5 as follows.

Theorem 2.6: A point process

has the UOSP if and only if for every n ≥ 1 such that P{T_n < ∞} = 1, the first n epochs times T₁ ≤ T₂ ≤ ··· ≤ T_n of N satisfy that

are distributed as the order statistics of n − 1 independent uniform(0,1) random variables.

3. SOME COMMENTS ON THE GENERAL OSP

In this section we extend some of the UOSP results in Section 2 to the more general order statistics property (OSP) that will be defined shortly. These extensions entail generalizations of the notions of [ell ]_∞-spherical and [ell ]_∞^≤-spherical densities.

Let a : [0,∞) → [0,∞) be a differentiable, strictly increasing function such that a(0) = 0. This function will be fixed throughout this section. Denote by a⁻¹ the inverse of a, and by a′, its derivative. If X₁,X₂,…,X_n have the joint density given in (1.1), then

has the density given by

which is a generalization of (1.1). The density function

of the corresponding order statistics is given by

which is a generalization of (1.2).

We now define the order statistics property (OSP), which is a generalization of the UOSP. Let {F_t}_t≥0 be a family of distribution functions such that F_t has the support [0,t] . We will say that a (finite or infinite) sequence of nonnegative random variables

has the OSP with respect to {F_t}_t≥0 if for any k ≥ 1 and for any t > 0 such that

, the conditional distribution of

, given that

, is the joint distribution of k order statistics based on a sample from F_t. Furthermore, consider a point process

with right-continuous sample paths, such that almost surely

for all

. Denote by

the jump times of the process. The process

is said to have the OSP with respect to {F_t}_t≥0 if for every

, such that

, the conditional distribution of

, given that

, is the joint distribution of k order statistics based on a sample from F_t.

Let

be a point process with right-continuous sample paths, such that almost surely

for all t ∈ [0,∞). Denote

, and γ = lim_t→∞ m(t). From Puri [15] it follows that if the process

has the OSP with respect to some {F_t}_t≥0 and γ = ∞, then

has to be a nonstationary mixed Poisson process; that is,

where K is a unit rate homogeneous Poisson process and W is a positive random variable, independent of K. On the other hand, if γ < ∞, then

has to be a mixed sample process with the kernel function

that is,

can be represented as the empirical distribution of a sample of a random size from the above distribution F.

Using the transformation (3.1) or its inverse, it is possible to obtain results about the OSP from results about the UOSP and vice versa. For example, if

have a joint density function

of the form (3.2), then any k-dimensional marginal of

is also of that form, k ≤ n (this statement is a generalization of Proposition 2.1). A generalization of Proposition 2.3 is the following.

Proposition 3.1: The random variables

have a density of the form (3.3) if and only if they have the OSP with respect to {F_t}_t≥0, where

From Proposition 3.1 we get the following generalization of Theorem 2.5.

Theorem 3.2: Let a : [0,∞) → [0,∞) be a differentiable, strictly increasing function such that a(0) = 0, and for t ≥ 0, let F_t be defined by (3.4). A point process

has the OSP with respect to {F_t}_t≥0 if and only if for every n ≥ 1 such that

, its first n epoch times

have a joint density of the form (3.3).

Example 3.3: The linear birth (Yule) process N = {N(t), t ≥ 0}, with initial population size k, is a nonhomogeneous pure birth process (see (1.4)–(1.6)) with intensity functions

where λ > 0 is some constant. Let T₁ ≤ T₂ ≤ ··· ≤ T_n be the first n jump times of N. The joint density function of T₁,T₂,…,T_n, evaluated at x₁ ≤ x₂ ≤ ··· ≤ x_n, can be computed as follows (see, e.g., (2.5) in Shaked et al. [16]; there is there an easily correctable mistake in the indexing):

where x₀ ≡ 0. A straightforward algebra yields (for x₁ ≤ x₂ ≤ ··· ≤ x_n)

that is, f_T1,T2,…,Tn is of the form (3.3) with

. Now, from Proposition 3.1 it follows that N has the OSP with respect to {F_t}_t≥0, where

Applying the inverse of the transformation (3.1), it follows that e^λT₁ − 1,e^λT₂ − 1,… have the UOSP. The latter result was obtained in Neuts and Resnick [14] by means of a straightforward, but lengthy, computation.

An interesting application in Bayesian inference in which densities of the form (3.2) arise naturally is described next. Let Θ = (Θ₁,Θ₂,…,Θ_n) be a vector of independent and identically distributed positive parameters, and let V (which may be a random variable or a random vector) be an observation whose distribution depends on Θ only through max_1≤i≤n Θ_i; that is, suppose that the joint density of Θ is of the form

where h is some density function on (0,∞), and that the conditional density of V given Θ = (θ₁,θ₂,…,θ_n) is of the form

for some function φ. Then, by Bayes' formula, the conditional density of Θ, given V = v, is given by

that is, given V = v, this density is of form (3.2). A particular application in reliability theory is described in the following example.

Example 3.4: Consider r components D₁,D₂,…,D_r whose lifetimes are denoted by V₁,V₂,…,V_r, respectively. Suppose that a parallel system of n identical components, with independent lifetimes Θ₁,Θ₂,…,Θ_n, supports the performance of D₁,D₂,…,D_r; that is, suppose that when the supporting parallel system is up, then the failure rate of each D_j is λ₀, whereas when the supporting parallel system is down, the failure rate of each D_j is λ₁, where λ₀ < λ₁. Assume that the joint density of Θ₁,Θ₂,…,Θ_n is given by (3.5). Then, the conditional density of (Θ₁,Θ₂,…,Θ_n), given (V₁,V₂,…,V_r), is of the form (3.6). It follows from Proposition 3.1 that the conditional distribution of the ordered Θ_i's, given (V₁,V₂,…,V_r), has the OSP with respect to {F_t}_t≥0, where

(H is the distribution function associated with h).

4. DISCRETE UOSP AND DISCRETE [ell ]_∞-SPHERICAL DENSITIES

Inspired by the UOSP for processes with continuous interepoch intervals, Huang and Shoung [9] have introduced a corresponding property for discrete-time point processes (i.e., processes with discrete interepoch intervals and unit step jumps). In this section, we will describe this property and a modification of it that is useful in some applications. We will then derive some discrete analogs of the results in Section 2. It turns out that in the discrete case, it is also of interest to consider processes that can have jumps larger than unity at the jump epochs; thus, we will also obtain further results that have no continuous analogs.

First, we introduce some notation and terminology that are used in this section and point out some simple observations. In this section, we use the notation

for {1,2,…}.

Let Z₁,Z₂,…,Z_n be

-valued random variables and set

Thus, X₁,X₂,…,X_n are also

-valued and they satisfy

If we denote by p_Z and p_X the discrete densities of (Z₁,Z₂,…,Z_n) and (X₁,X₂,…,X_n), respectively, then, obviously,

If, on the other hand, Z₁,Z₂,…,Z_n are

-valued random variables and X₁,X₂, …,X_n are as defined in (4.1), then X₁,X₂,…,X_n are also

-valued and they satisfy

In this case, the discrete density functions of (Z₁,Z₂,…,Z_n) and (X₁,X₂,…,X_n) satisfy

We will say that a discrete joint density p on (

[respectively

] is Schur-constant if

for some function φ, and that it is [ell ]_∞-spherical if

for some function φ. A discrete joint density p on

will be called [ell ]_∞^≤-spherical if

for some function ψ. A discrete joint density p on

will be called [ell ]_∞^<-spherical if

for some function ψ. Obviously, the above p_Z is Schur-constant on

[respectively

] if and only if p_X is [ell ]_∞^≤-spherical [respectively [ell ]_∞^<-spherical].

The following result is a discrete analog of Proposition 2.1. Its proof is a straightforward modification of the proof of Proposition 2.1 and is therefore omitted.

Proposition 4.1: If the discrete random variables X₁,X₂,…,X_n have an [ell ]_∞-spherical density f on

[respectively

], then any k-dimensional marginal of f is an [ell ]_∞-spherical density on

[respectively

], k ≤ n.

We will now define the UOSP for a (finite or infinite) sequence of discrete nonnegative random variables X₁,X₂,…. Two different definitions are needed depending on whether the X_i's can or cannot be equal to each other. If P{0 < X₁ < X₂ < ···} = 1, then we say that they have the UOSP (<) if for discrete 0 < x₁ < x₂ < ··· < x_k ≤ t we have

that is, conditional on X_k ≤ t and X_k+1 > t, the random variables X₁ < X₂ < ··· < X_k are distributed as the order statistics of a sample of size k drawn from the set {1,2,…,t} without replacement. If, on the other hand, P{0 ≤ X₁ ≤ X₂ ≤ ···} = 1, then we say that these X_i's have the UOSP (≤) if for discrete 0 ≤ x₁ ≤ x₂ ≤ ··· ≤ x_k ≤ t we have

that is, conditional on X_k ≤ t and X_k+1 > t, the random variables X₁ ≤ X₂ ≤ ··· ≤ X_k are distributed as the order statistics of a sample of size k drawn from the set {0,1,…,t} with replacement. (The latter claim follows from the fact that

is the number of ways that k objects can be put into t + 1 cells; see, e.g., Feller [6, p.38].)

The following result is a discrete analog of Proposition 2.3. Again we omit the proof, as it is a straightforward modification of the proof of Proposition 2.3.

Proposition 4.2: The discrete random variables 0 < X₁ < X₂ < ··· < X_n [respectively 0 ≤ X₁ ≤ X₂ ≤ ··· ≤ X_n] have an [ell ]_∞^<-spherical density as in (4.4) [respectively [ell ]_∞^≤-spherical density as in (4.3)] if and only if they have the UOSP (<) [respectively UOSP (≤)].

An analog of Proposition 2.4 can also be stated and proven.

In the following, we will now study two kinds of discrete-time discrete-state process: those that have only unit step jumps and those that are more general. We start with discrete-time point processes that have only unit step jumps. Let {A(t), t = 0,1,…} be such a process with A(0) ≡ 0. We assume that P{lim_t→∞ A(t) = ∞} = 1. The distributional properties of {A(t), t = 0,1,…} can be equivalently described by means of either one of three sequences of random variables:

A process {A(t), t = 0,1,…} as above will be said to have the UOSP (<) if T₁,T₂,… have this property.

We now recall a definition and a result from Huang and Shoung [9].

Definition 4.3 [9, p.72]: A discrete-time discrete-state process {A(t), t = 0,1,…} is called a mixed geometric process if there exists a random variable Θ that takes on values in (0,1) such that, given that Θ = θ, the interepoch intervals Z₁,Z₂,… of {A(t), t = 0,1,…} are independent and identically distributed with probabilities P_θ{Z_k = z} = θ(1 − θ)^z−1, z ≥ 1, k ≥ 1.

Theorem 4.4 [9, p.73]: A discrete-time discrete-state process {A(t), t = 0,1,…} with unit step jumps, such that A(0) = 0 and P{lim_t→∞ A(t) = ∞} = 1, has the UOSP (<) if and only if it is a mixed geometric process.

We can now obtain the following result; part of it is a discrete analog of Theorem 2.5.

Theorem 4.5: Let {A(t), t = 0,1,…} be a discrete-time discrete-state process with unit step jumps such that A(0) = 0 and P{lim_t→∞ A(t) = ∞} = 1. The following statements are equivalent:

(iii) The interepoch intervals Z₁,Z₂,…,Z_k have a Schur-constant density on

for every k ≥ 1.

(iv) The epoch times T₁,T₂,…,T_k have a joint [ell ]_∞^<-spherical density on

for every k ≥ 1.

Proof: The equivalence (i) [hArr ] (ii) is Theorem 4.4. The equivalence (i) [hArr ] (iv) follows from Proposition 4.2. The equivalence (iii) [hArr ] (iv) follows from the comment after (4.4).

(i) ⇒ (v): Fix a t ≥ 1. Let (i₁,i₂,…,i_t) ∈ {0,1}^t. Suppose that k of the i_j's are one and that the other t − k indicators are zero. Then, for some 0 < t₁ < t₂ < ··· < t_k ≤ t, we have

So,

where the second equality follows from the assumption that T₁,T₂,…,T_k have the UOSP (<); that is, P{I₁ = i₁,I₂ = i₂,…,I_t = i_t} does not depend on the actual values i₁,i₂,…,i_t, but only on the related k. Thus, I₁,I₂,…,I_t are exchangeable.

(v) ⇒ (ii): From the de Finetti theorem there exists a probability measure Π on (0,1) such that

for any t ≥ 1, where k is the number of i_j's that are one.

Now, fix a k ≥ 1 and a

. Denote

. Then, from (4.5) we get

that is, the process {A(t), t = 0,1,…} is a mixed geometric process. █

The equivalence (ii) [hArr ] (iii) is a well-known de Finetti-type theorem (see, for example, Diaconis and Freedman [5, p.414]). Also, it can be viewed as a discrete analog of Theorem 1.44 in Spizzichino [17].

Using Theorem 4.5, one can obtain interesting discrete Schur-constant, [ell ]_∞^<-spherical, and exchangeable densities. The following example illustrates this.

Example 4.6: Let Θ in Definition 4.3 have the beta density with parameters α > 0 and β > 0. Then, the Schur-constant joint discrete density of the first k ≥ 1 interepoch intervals is given by

This density (actually the density of (Z₁ − 1,Z₂ − 1,…,Z_k − 1)) is a special case of the beta-compound negative multinomial density (see Johnson, Kotz, and Balakrishnan [11, p.104]); see also Example 4.9. The marginal density of Z₁ (which, by exchangeability, is the marginal density of each Z_i) is given by

This density (actually the density of Z₁ − 1) is a special case of the beta-negative-binomial density, which is also called the Markov–Pólya density; see Johnson, Kotz, and Kemp [12, pp.242,243], but note the different notation; for example, for a > 1, they denote (−a)! = 1/(a − 1)!, and for any nonzero a and b, they denote

, where for a > 0, they use the notation a! = Γ(a + 1).

The [ell ]_∞^<-spherical joint discrete density of the first k ≥ 1 epoch times is given by

Finally, the exchangeable joint density of the first t ≥ 1 indicators can be computed as follows:

Marginally, each I_j is a Bernoulli random variable with parameter α/(α + β).

In the case when Θ is a uniform(0,1) random variable (i.e., α = β = 1), we have

in particular,

The [ell ]_∞^<-spherical joint discrete density of the first k ≥ 1 epoch times is then

and the exchangeable density of the first t ≥ 1 indicators is then

Marginally, each I_j is then a Bernoulli random variable with parameter ½.

We now proceed to the study of nondecreasing discrete-time discrete-state processes that can have jumps larger than unity at the jump epochs. These can be described by allowing the interepoch intervals between the “unit jumps” to be equal to zero with a positive probability so that an actual jump of size j > 0 corresponds to a sequence of j consecutive interepoch intervals of size 0. Thus, let {B(t), t = 0,1,…} be such a process (here, we allow jumps at time 0, so we do not require that B(0) = 0 here). The distributional properties of {B(t), t = 0,1,…} can be equivalently described by means of either one of three sequences of random variables:

[bull ] The interepoch intervals between “unit jumps” Z₁,Z₂,…; that is,

(here, the Z_i's can be equal to zero with positive probability).

[bull ] The times of the “unit jumps” T₁,T₂,…; that is,

(here consecutive T_k's may be equal to each other with positive probability).

A process {B(t), t = 0,1,…}, as above, will be said to have the UOSP (≤) if T₁,T₂,… have this property.

A process {B(t), t = 0,1,…}, as above (i.e., which can have jumps larger than unity), can be a discrete-time discrete-state mixed uniform sample process. In order to define it formally, let

be a positive integer and let U₁,U₂,… be a sequence of independent random variables distributed uniformly on

. Let L be a positive integer-valued random variable, independent of the U_i's. Given L = l, let

be the empirical distribution function based on U₁,U₂,…,U_l. The unconditional process

will be called a discrete-time discrete-state mixed uniform sample process. Such processes are useful in studies where the discrete uniform distribution can model some real phenomenon, and a sample, of a size that cannot be predetermined in advance of the experiment, is taken from that distribution.

For processes, as above, that can have general (rather than unit) jumps, we have the following equivalent conditions.

Theorem 4.7: Let {B(t), t = 0,1,…} be a nondecreasing discrete-time discrete-state process. The following statements are equivalent:

Proof: The equivalence (i) [hArr ] (iv) follows from Proposition 4.2. The equivalence (iii) [hArr ] (iv) follows from the comment after (4.4).

(i) ⇒ (ii): The proof of this statement is a modification of the proof of Theorem 4 of Puri [15]. Fix an integer

and set

. From the assumption that {B(t), t = 0,1,…} has the UOSP (≤) it follows that, given L = l, the jump times

are distributed as the order statistics of a sample of size l from a uniform distribution on

; that is,

is an empirical distribution function corresponding to a sample of size l from a uniform distribution on

.

(ii) ⇒ (i): Fix a

and let

. Given L = l, we have that

is an empirical distribution function corresponding to a sample of size l from a uniform distribution on

. Let T₁ ≤ T₂ ≤ ··· ≤ T_l be the jump points of this empirical distribution (some of the T_i's might be equal to each other if a jump is larger than unity). These T_i's obviously have the UOSP (≤).

(i) ⇒ (v): Fix a t ≥ 0. Let

be a possible realization of (J₀,J₁,…,J_t). Set

these are the corresponding realized epochs of actual jumps—obviously, there is a finite number k of them. Now we describe the corresponding realization of the epoch times t_m's of the zero and nonzero jumps.

Note that above we have described

epoch times t_m's; the second equality here follows from the fact that j_v's with indices different than α_u are all zeros. Let us order these t_m's as 0 ≤ t₁ ≤ t₂ ≤ ··· ≤ t_p (note that t_p ≤ t). Then,

So,

where the second equality follows from the assumption that T₁,T₂,…,T_p have the UOSP (≤); that is, P{J₀ = j₀,J₁ = j₁,…,J_t = j_t} does not depend on the actual values j₀,j₁,…,j_t, but only on

. Thus, J₀,J₁,…,J_t have a Schur-constant density.

(v) ⇒ (iv): Fix a k ≥ 1 and 0 ≤ t₁ ≤ t₂ ≤ ··· ≤ t_k. Let

that is, t_ε1 < t_ε2 < ··· < t_εh are the different values in the sequence t₁ ≤ t₂ ≤ ··· ≤ t_k. Obviously, h above is finite (it satisfies h ≤ k). Now, define t_k values j₀,j₁,…,j_tk−1 as follows: Let

and let the other t_k − h + 1 values j_m's be zero. Also, let

that is, a is the number of the last t_n's in the sequence t₁ ≤ t₂ ≤ ··· ≤ t_k that are equal to each other. Obviously, 1 ≤ a ≤ k. Then,

Note that

the last equality follows by noting that ε_h is the index of the (first) largest value in the sequence t₁ ≤ t₂ ≤ ··· ≤ t_k and the number of t_n's that are equal to t_εh is a. Therefore,

Since (J₀,J₁,…,J_tk) has a Schur-constant density on (

∪ {0})^t_k+1, it follows that there exists a function φ_tk such that

Thus, for a fixed k ≥ 1, when 0 ≤ t₁ ≤ t₂ ≤ ··· ≤ t_k, the probability P{T₁ = t₁, T₂ = t₂,…,T_k = t_k} depends only on t_k; that is, T₁,T₂,…,T_k have a joint [ell ]_∞^≤-spherical density on

. █

It is of interest to compare the statement in Theorem 4.5(v) with the one in Theorem 4.7(v). The condition on the jump variables in the former is exchangeability, whereas in the latter, the condition on the jump variables is that they have a Schur-constant density. However, in the setup of Theorem 4.5, these two conditions are the same because for binary random variables, Schur-constant density and exchangeability are equivalent conditions.

It is also of interest to compare the statement in Theorem 4.5(ii) with the one in Theorem 4.7(ii). It is seen that in the discrete-time case, the mixed geometric process is a natural tool for characterizing processes that have unit step jumps and are unbounded, whereas the mixed uniform sample process plays a similar role for characterizing processes that can be bounded and that can have jumps larger than unity.

Remark 4.8: An interesting relationship exists between processes {A(t), t = 0,1,…} such that P{lim_t→∞ A(t) = ∞} = 1 with the UOSP (<) and processes {B(t), t = 0,1,…} such that P{lim_t→∞ B(t) = ∞} = 1 with the UOSP (≤). Let Z₁,Z₂,… be the interepoch intervals of {A(t), t = 0,1,…} as in Theorem 4.5(iii). Define

and let the Z_i′'s be the interepoch intervals of a process {B(t), t = 0,1,…} as in Theorem 4.7(iii). If the Z₁,Z₂,…,Z_k have a Schur-constant density on

for each k ≥ 1, then so do Z₁′,Z₂′,…,Z_k′ on

, and vice versa. We thus see that the transformation Z_i → Z_i′ described above, and its inverse, enable us to go back and forth between processes that have the UOSP (<) and the UOSP (≤), provided they satisfy P{lim_t→∞ A(t) = ∞} = 1 and P{lim_t→∞ B(t) = ∞} = 1.

The transformation (4.9) indicates that unbounded processes with unit step jumps (as in Theorem 4.5) that satisfy any of the equivalent conditions in Theorem 4.5 also satisfy a version of the statement in Theorem 4.7(ii). A continuous-time analog of this observation is Theorem 6.6 in Grandell [7].

We will now obtain a relationship among discrete Schur-constant densities that is a discrete analog of (2.1). This will be used in the sequel. Let W₁,W₂,… be a sequence of random variables taking on values in

, such that the density function of (W₁,W₂,…,W_n) is of the form (4.2) where the function φ there is denoted here by φ_n, n = 1,2,…. Then, for any

, we have

that is,

Next, we express the joint density of the jump amounts J₀,J₁,…,J_t by means of the densities of the interepoch intervals Z_i's. For this purpose, let us assume that the density of (Z₁,Z₂,…,Z_n) is of the form (4.2) where the function φ there is denoted here by φ_n, n = 1,2,…. For t ≥ 0, fix a

and let

, then we have

On the other hand, if p ≥ 1, then

where the first equality follows from the fact that (J₀,J₁,…,J_t) has a Schur-constant density, the second equality essentially follows from (4.7), and the next to last equality follows from (4.10).

In a similar manner, we can express the joint density of the interepoch intervals Z₁,Z₂, …,Z_k by means of the densities of the jump amounts J_i's. For k ≥ 1, fix a

and let

. If w = 0, then we have

On the other hand, if w ≥ 1, then it can be shown, in a manner similar to the proof of (4.12) (but using (4.8) rather than (4.7)), that

Theorem 4.7 and the transformation (4.9), together with the above observations, enable us to obtain interesting Schur-constant densities on

. The following example illustrates this.

Example 4.9: Let Θ and the Z_i's be as in Example 4.6 and let Z_i′'s be defined by (4.9). Then, it is a simple matter to see that the Schur-constant density of the first k Z_i′'s is given by

This density is a special case of the beta-compound negative multinomial density (see Johnson et al. [11, p.104]).

In order to find the joint density of the first t + 1 ≥ 1 jumps amounts J₀,J₁,…,J_t, we use (4.12). Fix a

and denote

. If p ≥ 1, then, by (4.12) and (4.13), we have

If p = 0, then, by (4.11), the corresponding probability P{J₀ = 0,J₁ = 0,…,J_t = 0} is a sum of expressions of the form (4.6).