Abstract

We consider probability distributions with constant rate on partially ordered sets, generalizing distributions in the usual reliability setting that have constant failure rate. In spite of the minimal algebraic structure, there is a surprisingly rich theory, including moment results and results concerning ladder variables and point processes. We concentrate mostly on discrete posets, particularly posets whose graphs are rooted trees. We pose some questions on the existence of constant rate distributions for general discrete posets.

1. Preliminaries

1.1. Introduction

The exponential distribution on and the geometric distribution on are characterized by the constant rate property: the density function (with respect to Lebesgue measure in the first case and counting measure in the second) is a multiple of the upper (right-tail) distribution function.

The natural mathematical home for the constant rate property is a partially ordered set (poset) with a reference measure for the density functions. In this paper we explore these distributions. In spite of the minimal algebraic structure of a poset, there is a surprisingly rich theory, including moment results and results concerning ladder variables and point processes. In many respects, constant rate distributions lead to the most random way to put ordered points in the poset. We will be particularly interested in the existence question–-when does a poset support constant rate distributions?

1.2. Standard Posets

Suppose that is a poset. For , let

For and , let

For , is said to cover if is a minimal element of . If is countable, the Hasse graph or covering graph of has vertex set and (directed) edge set . We write if and are comparable, and we write if and are noncomparable. The poset is connected if, for every , there exists a finite sequence such that , , and for .

Now suppose that is a -algebra on and let denote the corresponding product -algebra on for . The main assumption that we make to connect the algebraic structure of to the measure structure is that the partial order is itself measurable, in the sense that . It then follows that for since these sets are simply the cross-sections of (see [1, 2]). Note that , so in fact all of the “intervals” , , and so forth are measurable for . Also, for and . If is countable, is the power set of . When is uncountable, is usually the Borel -algebra associated with an underlying topology (see [3, 4]).

Finally, we fix a positive, -finite measure on as a reference measure. We assume that and for each . When is countable, we take to be counting measure on unless otherwise noted. In this case, must be finite for each , so that is locally finite in the terminology of discrete posets [5].

Definition 1.1. The term standard poset will refer to a poset together with a measure space that satisfies the algebraic and measure theoretic assumptions. When the measure space is understood (in particular for discrete posets with counting measure), we will often omit the reference to this space.
An important special case is the poset associated with a positive semigroup. A positive semigroup is a semigroup that has an identity element , satisfies the left-cancellation law, and has no nontrivial inverses. The partial order associated with is given by If , then satisfying is unique and is denoted by . The space has a topology that makes the mapping continuous, and is the ordinary Borel -algebra. The reference measure is left invariant:
Probability distributions (particularly exponential type distributions) on positive semigroups have been studied in [6–10]. The critical feature of a positive semigroup is that, for each , looks like the entire space from algebraic, topological, and measure theoretic points of view. Of course, there is no such self-similarity for general standard posets. Nonetheless, the definition of a standard poset is supposed to capture the minimal structure of a temporal (i.e., time-like) set, particularly in probability applications such as reliability and queuing. The following examples are intended to give some idea of the structures that we have in mind; some of these will be explored in more detail in this paper.

Example 1.2. Consider that , where is ordinary order (and where is Lebesgue measure on the Borel -algebra ) is the standard model of continuous time. Of course, this poset is associated with the positive semigroup .

Example 1.3. Consider that , where is ordinary order, is the standard model of discrete time. This poset is associated with the positive semigroup .

Example 1.4. If is a standard poset for , then so is where is the Cartesian product, the product order, the product -algebra, and the product measure. Product sets occur frequently in multivariate settings of reliability and other applications. In particular, the -fold powers of the posets in Examples 1.2 and 1.3 are the standard multivariate models for continuous and discrete time, respectively.

Example 1.5. Consider that , where means that divides , is a standard poset and corresponds to the positive semigroup . Exponential distributions for this semigroup were explored in [10].

Example 1.6. Suppose that is a poset and that is a poset for each . The lexicographic sum of over is the poset where and where if and only if either or and . In the special case that for each , is the lexicographic product of and . In the special case that is the equality relation, is the simple sum of over .
Given appropriate -algebras and reference measures, these become standard posets and have possible applications. For example, suppose that is a poset in a reliability model, for each . The simple sum could represent the appropriate space when devices are run in parallel. On the other hand, sometimes the more exotic posets lead to additional insights into simple posets. For example, is isomorphic to the lexicographic product of with (just consider the integer part and remainder of ).

Example 1.7. Suppose that is a standard poset and that with for each . Then is also a standard poset, where is the restriction of to and is the restriction of to . An incredibly rich variety of new posets can be constructed from a given poset in this way. Moreover, such subposets can be used to impose restrictions in the reliability setting and other applications.

Example 1.8. A rooted tree is the Hasse graph of a standard poset, and such trees have myriad applications to various kinds of data structures. One common application is to reproducing objects (e.g., organisms in a colony or electrons in a multiplier). In this interpretation, that covers means that is a child of , and, more generally, means that is an ancestor of .

Example 1.9. If denotes the set of all finite subsets of , then is a standard discrete poset. Somewhat surprisingly, this poset corresponds to a positive semigroup. Exponential type distributions for this semigroup were explored in [9]. Random sets have numerous applications (see [11]); in particular, the problem of choosing a subset of in the “most random” way is important in statistics.

Example 1.10. Various collections of finite graphs can be made into standard posets under the subgraph relation. Depending on the model of interest, the graphs could be labeled or unlabeled and might be restricted (e.g., to finite rooted trees). Of course, random graphs in general and random trees in particular are large and important areas of research (see [12, 13]).

For the remainder of this paper, unless otherwise noted, we assume that is a standard poset.

1.3. Operators and Cumulative Functions

Let denote the set of measurable functions from into that are bounded on for each . Define the lower operator on by Next, let denote the usual Banach space of measurable functions , with . Define the upper operator on by A simple application of Fubini's theorem gives the following duality relationship between the linear operators and : assuming, of course, the appropriate integrability conditions. Both operators can be written as integral operators with a kernel function. Define by Then In the discrete case, is the Riemann function in the terminology of Möbius inversion [5], and its inverse (also in the sense of this theory) is the Möbius function. The lower operator is invertible, and if then As we will see in Example 2.2, the upper operator is not invertible in general, even in the discrete case.

Now let 1 denote the constant function on , and define by 1 for . Equivalently, for and , where is -fold product measure on . We will refer to as the cumulative function of order ; these functions play an important role in the study of probability distribution on . For the poset of Example 1.2, the cumulative functions are For the poset of Example 1.3, the cumulative functions are For the poset of Example 1.5, is the number of factorings of ; these are important functions in number theory. For the poset of Example 1.9, the cumulative functions are

The ordinary generating function of is the function given by for and for for which the series converges absolutely. The generating function arises in the study of the point process associated with a constant rate distribution. For the poset of Example 1.2, . For the poset of Example 1.3,

2. Probability Distributions

2.1. Distribution Functions

Suppose now that is a random variable taking values in . We assume that if and only if for . Thus, the distribution of is absolutely continuous with respect to , and the support of is the entire space in a sense. Let denote the probability density function (PDF) of , with respect to .

Definition 2.1. The upper probability function (UPF) of is the function given by
If is interpreted as a temporal space and as the failure time of a device, then is the reliability function; is the probability that the failure of the device occurs at or after (in the sense of the partial order). This is the usual meaning of the reliability function in the standard spaces of Examples 1.2 and 1.3 and of the multivariate reliability function in the setting of Example 1.4 (see [14]). The definition is relevant for more exotic posets as well, such as the lexicographic order discussed in Example 1.6. Note that In general, the UPF of does not uniquely determine the distribution of .

Example 2.2. Let be fixed set with elements (), and let denote the lexicographic sum of the antichains over , where and for . Let be a PDF on with UPF . Define by where is a constant. It is straightforward to show that In particular, . Hence if we can choose a PDF and a nonzero constant so that for every , then is a PDF different from but with the same UPF. This can always be done when .

In the discrete case, a simple application of the inclusion-exclusion rule shows that the distribution of is determined by the generalized UPF, defined for finite by An interesting problem is to give conditions on the poset that ensure that a distribution on is determined by its ordinary UPF. This holds for a discrete upper semilattice, since for finite. It also holds for trees, as we will see in Section 5.

The following proposition gives a simple result that relates expected value to the lower operator . For positive semigroups, this result was given in [10].

Proposition 2.3. Suppose that has UPF . For and ,

Proof. Using Fubini's theorem,

When has a PDF , (2.6) also follows from (1.7). In particular, letting gives and when , (2.8) becomes For the poset of Example 1.2, (2.8) becomes and (2.9) reduces to the standard result . For the poset of Example 1.3, (2.8) becomes and (2.9) reduces to the standard result . For the poset of Example 1.9, (2.8) becomes

Definition 2.4. Suppose that has UPF and PDF . The rate function of is the function defined by
Again, if is interpreted as a temporal space and as the failure time of a device, then is the failure rate function: gives the probability density of failure at given that failure occurs at or after . As mentioned before, this agrees with the usual meaning of failure rate in the standard spaces of Examples 1.2 and 1.3 and with the usual multivariate meaning for the product of these spaces (see [14]).
In general, has constant rate if is constant on , increasing rate if is increasing on and decreasing rate if is decreasing on . In the reliability setting, devices with increasing failure rate deteriorate over time, in a sense, while devices with decreasing failure rate improve. Devices with constant failure rate do not age, in a sense.

Proposition 2.5. Suppose that is a standard discrete poset and that has rate function . Then for and if and only if is maximal.

Proof. Let denote the PDF of and the UPF. For , Hence By (2.15), for all . If is maximal then so . Conversely, if is not maximal, then (since has support ) and hence .

2.2. Ladder Variables and Partial Products

Let be a sequence of independent, identically distributed random variables, taking values in , with common UPF and PDF . We define the sequence of ladder variables associated with as follows. First let and then recursively define

Proposition 2.6. The sequence is a homogeneous Markov chain with transition density given by

Proof. Let . The conditional distribution of given corresponds to observing independent copies of until a variable occurs with a value greater than (in the partial order). The distribution of this last variable is the same as the conditional distribution of given , which has density on .

Since has PDF , it follows immediately from Proposition 2.6 that has PDF (with respect to ) given by This PDF has a simple representation in terms of the rate function :

Suppose now that is a positive semigroup and that is an IID sequence in with PDF . Let for , so that is the partial product sequence associated with

Proposition 2.7. The sequence is a homogeneous Markov chain with transition probability density given by

Since has PDF , it follows immediately from Proposition 2.7 that has PDF (with respect to ) given by So in the case of a positive semigroup, there are two natural processes associated with an IID sequence : the sequence of ladder variables and the partial product sequence . In general, these sequences are not equivalent but, as we will see in Section 3, are equivalent when the underlying distribution of has constant rate. Moreover, this equivalence characterizes constant rate distributions.

Return now to the setting of a standard poset. If is an increasing sequence of random variables in (such as a sequence of ladder variables or, in the special case of a positive semigroup, a partial product sequence), we can construct a point process in the usual way. For , let so that is the number of random points in . We have the usual inverse relation between the processes and , namely, if and only if for and . For , let denote the lower probability function of , so that for . Then for . Of course, . If we define for all , then for fixed , is the upper probability function of .

The sequence of random points can be thinned in the usual way. Specifically, suppose that each point is accepted with probability and rejected with probability , independently from point to point. Then the first accepted point is where is independent of and has the geometric distribution on with parameter .

3. Distributions with Constant Rate

3.1. Characterizations and Properties

Suppose that is a random variable taking values in with UPF . Recall that has constant rate if is a PDF of .

If is associated with a positive semigroup , then has an exponential distribution if Equivalently, the conditional distribution of given by is the same as the distribution of . Exponential distributions on positive semigroups are studied in [6–10]. In particular, it is shown in [7] that a distribution is exponential if and only if it has constant rate and However there are constant rate distributions that are not exponential. Moreover, of course, the exponential property makes no sense for a general poset.

Constant rate distributions can be characterized in terms of the upper operator or the lower operator . In the first case, the characterization is an eigenvalue condition and in the second case a moment condition.

Proposition 3.1. The poset supports a distribution with constant rate if and only if there exists a strictly positive with

Proof. If is the UPF of a distribution with constant rate , then trivially satisfies the conditions of the proposition since is a PDF with UPF . Conversely, if is strictly positive and satisfies (3.3), then is a PDF and , so the distribution with PDF has constant rate .

Proposition 3.2. Random variable has constant rate if and only if for every .

Proof. Suppose that has constant rate and UPF , so that is a PDF of . Let From Proposition 2.3, Conversely, suppose that (3.4) holds for every . Again let denote the UPF of . By Proposition 2.3, condition (3.4) is equivalent to It follows that is a PDF of .

If has constant rate , then iterating (3.4) gives In particular, if then and if , .

Suppose now that is a discrete standard poset and that has constant rate on . From Proposition 2.5, , and if , all elements of are maximal, so that is an antichain. Conversely, if is an antichain, then any distribution on has constant rate 1. On the other hand, if is not an antichain, then and has no maximal elements.

Consider again a discrete standard poset . For , we say that and are upper equivalent if . Upper equivalence is the equivalence relation associated with the function from to . Suppose that has constant rate on and UPF . If are upper equivalent, then so from (2.15), Thus, the UPF (and hence also the PDF) is constant on the equivalence classes.