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- Table of Contents
Journal of Probability and Statistics
Volume 2010 (2010), Article ID 675754, 21 pages
Constant Rate Distributions on Partially Ordered Sets
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA
Received 5 April 2010; Revised 2 August 2010; Accepted 30 August 2010
Academic Editor: Rongling Wu
Copyright © 2010 Kyle Siegrist. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 .
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 .
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 . Random sets have numerous applications (see ); 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 , 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 ). 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 .
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 ).
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  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.
For the poset in Example 1.2, of course, the constant rate distributions are the ordinary exponential distributions (see  for myriad characterizations). For the poset in Example 1.3, the constant rate distributions are the ordinary geometric distributions. For the poset as described in Example 1.4, the constant rate distributions are mixtures of distributions that correspond to independent, exponentially distributed coordinates .
3.2. Ladder Variables and Partial Products
Assume that is a standard poset. Suppose that is an IID sequence with common UPF , and let be the corresponding sequence of ladder variables.
Proposition 3.3. If the distribution has constant rate then (1) is a homogeneous Markov chain on with transition probability density given by (2) has PDF given by has PDF given by the conditional distribution of given is uniform on
Proof. Parts 1 and 2 follow immediately from Proposition 2.6. Part 3 follows from Part 2 and Part 4 from Parts 2 and 3.
Part 4 shows that the sequence of ladder variables of an IID constant rate sequence is the most random way to put a sequence of ordered points in . In the context of Example 1.2, of course, is the sequence of arrival times of an ordinary Poisson process and the distribution in Part 3 is the ordinary gamma distribution of order and rate . In the context of Example 1.3, the distribution in Part 3 is the negative binomial distribution of order and parameter . Part 4 almost characterizes constant rate distributions.
Proposition 3.4. Suppose that is a standard, connected poset and that is the sequence of ladder variables associated with an IID sequence . If the conditional distribution of given is uniform on , then the common distribution of has constant rate.
Proof. From (2.20), the conditional PDF of given is where is the normalizing constant. But this is constant in by assumption, and hence is constant on for each . Thus, it follows that whenever . Since is connected, is constant on .
If the poset is not connected, it is easy to construct a counterexample to Proposition 3.4. Consider the simple sum of two copies of . Put proportion of a geometric distribution with rate on the first copy and of a geometric distribution with rate on the second copy, where and . The resulting distribution has rate on the first copy and rate on the second copy. If is an IID sequence with the distribution so constructed and the corresponding sequence of ladder variables, then the conditional distribution of given is uniform for each .
Suppose now that is an IID sequence from a distribution with constant rate and UPF . Let denote the corresponding sequence of ladder variables, and suppose that the sequence is thinned with probability as described in Section 2.2. Let denote the first accepted point.
Proposition 3.5. The PDF of is given by where is the generating function associated with .
Proof. For ,
In general, does not have constant rate but, as we will see in Section 5, does have constant rate when is a tree.
Suppose now that is a positive semigroup and that is an IID sequence. Let denote the sequence of ladder variables and the partial product sequence. If the underlying distribution of is exponential, then the distribution has constant rate, so Proposition 3.3 applies. But by Proposition 2.7, is also a homogeneous Markov chain with transition probability where is the rate constant, the PDF, and the UPF. Thus also satisfies the results in Proposition 3.3, and, in particular, and are equivalent. The converse is also true.
Proposition 3.6. If and are equivalent, then the underlying distribution of is exponential.
Proof. Let denote the common PDF of and the corresponding UPF. Since , the equivalence of and means that the two Markov chains have the same transition probability density, almost surely with respect to . Thus we may assume that Equivalently, Letting we have , so the distribution has constant rate . But then we also have , so , and hence the distribution is exponential.
4. Relative Aging
In most cases, the reference measure of a standard poset is natural in some sense-counting measure for discrete posets, Lebesgue measure for Euclidean posets, for example. However, another possibility is to use a given probability distribution on to construct a reference measure and then to study the rate functions of other probability distributions relative to this reference measure. In this way, we can study the “relative aging” of one probability distribution with respect to another. (This was done for positive semigroups in .)
Thus, suppose that is a poset and that is a -algebra of subsets of satisfying the assumptions in Section 1.2. Suppose that is a random variable taking values in , with UPF assumed to be strictly positive. Let denote the distribution of , so that for . If we take itself to be the reference measure, then, trivially of course, the PDF of is the constant function 1 (so that is “uniformly distributed” on with respect to ). Thus the rate function is , so that, curiously, has increasing rate with respect to its own distribution.
Of course, this is not quite what we want. We would like to construct a measure on that gives constant rate 1 and then study the rate of other distributions relative to this measure. Define the measure on by Thus, So has PDF , and hence has constant rate , with respect to . Now if is another random variable with values in , then the definitions and results of Sections 2 and 3 apply, relative to . It is easy to see that has rate function with respect to , then has rate function with respect to . In particular, has increasing (decreasing) (constant) rate with respect to if and only if has decreasing (increasing) (constant) rate with respect to , respectively.
Finally, suppose that does in fact have a natural reference measure , so that is a standard poset. If and are random variables with rate functions and (with respect to ), respectively, then the rate function of with respect to is .
In this section we consider a standard discrete poset whose covering graph is a rooted tree. Aside from the many applications of rooted trees, this is one of the few classes of posets for which explicit computations are possible.
The root is the minimum element. When , there is a unique path from to and we let denote the distance from to . We abbreviate by . Let denote the children of , and more generally let for and . Thus, and partitions . When , we write instead of .
The only trees that correspond to positive semigroups are those for which has the same cardinality for every . In this case, the poset is isomorphic to the free semigroup on a countable alphabet, with concatenation as the semigroup operation .
5.1. Upper Probability Functions
Let be a random variable with values in having PDF and UPF . Then In particular, uniquely determines . Moreover, we can characterize upper probability functions.
Proposition 5.1. Suppose that . Then is the UPF of a distribution on if and only if(1),(2) for every ,(3) as .
Proof. Suppose first that is the UPF of a random variable taking values in . Then trivially , and, by (5.4),
Next, if and only if for some . Moreover the events are disjoint over . Thus
But by local finiteness, the random variable (taking values in ) has a proper (nondefective) distribution, so as .
Conversely, suppose that satisfies conditions (1.7)–(1.15). Define on by Then for by (1.12). Suppose that and let . Then since and since the sum collapses. But Thus letting in (5.8), we have Letting in (5.10) gives so is a PDF on . Another application of (5.10) then shows that is the UPF of .
Note that is a lower semilattice. Hence if and are independent random variables with values in , with UPFs and , respectively, then has UPF .
Proposition 5.2. Suppose that satisfies and For , define by . Then is an UPF on .
Note that is not itself an UPF, unless the tree is a path, since properties and in Proposition 5.1 will fail in general. Thus, even when is an UPF, we cannot view simply as the product of two UPFs in general. However, we can give a probabilistic interpretation of the construction in Proposition 5.2 in this case. Thus, suppose that is a random variable taking values in with UPF and PDF . Moreover, suppose that each edge in the tree , independently of the other edges, is either working with probability or failed with probability . Define by
Corollary 5.3. Random variable has UPF given in Proposition 5.2.
Proof. If and , then if and only if the path from to is working. Hence for and . So conditioning on gives
5.2. Rate Functions
Next we are interested in characterizing rate functions of distributions that have support . If is such a function, then, from Proposition 2.5, and if and only if is a leaf. Moreover, if is the UPF, then and Conversely, these conditions give a recursive procedure for constructing an UPF corresponding to a given rate function. Specifically, suppose that and that for every leaf . First, we define . Then if has been defined for some and is not a leaf, then we define for arbitrarily, subject only to the requirement that and that (5.16) holds. Note that satisfies the first two conditions in Proposition 5.1. Hence if satisfies the third condition, then is the UPF of a distribution with support and with the given rate function . The following proposition gives a simple sufficient condition.
Proposition 5.4. Suppose that and that for each leaf . If there exists such that for all , then is the rate function of a distribution with support .
Proof. Let be any function constructed according to the recursive procedure above. Then as noted above, satisfies the first two conditions in Proposition 5.1. A simple induction on shows that so the third condition in Proposition 5.1 holds as well.
Condition (5.17) means that the distribution of is stochastically smaller than the geometric distribution on with rate constant . If is not a path, then the rate function does not uniquely determine the distribution. Indeed, if has two or more children, then there are infinitely many ways to satisfy (5.16) given .
5.3. Constant Rate Distributions
Recall that if has maximal elements (leaves), then there are no constant rate distribution with support , except in the trivial case that is an antichain.
Corollary 5.5. Suppose that is a rooted tree without leaves. Then is the UPF of a distribution with constant rate if and only if and
Proof. This follows immediately from Proposition 5.4.
Corollary 5.6. Suppose that has constant rate on . Then has the geometric distribution on with rate .
Proof. For ,
As a special case of the comments in Section 5.2, we can construct the UPFs of constant rate distributions on recursively: start with . If is defined for a given , then define for arbitrarily, subject only to the conditions and that (5.18) holds.
5.4. Ladder Variables and the Point Process
Let be the UPF of a distribution with constant rate , so that satisfies the conditions in Corollary 5.5. Let be an IID sequence with common distribution and let be the corresponding sequence of ladder variables. By Proposition 3.3, the distribution of has PDF
Consider now the thinned point process associated with where a point is accepted with probability and rejected with probability , independently from point to point.
Proposition 5.7. The distribution of the first accepted point has constant rate .
Proof. By Proposition 3.5, the PDF of the first accepted point is Consider the function given by Note that and for
In Proposition 5.7, the UPF is related to the UPF by the construction in Corollary 5.3. That is, suppose that denotes the first accepted point in the thinned process. Then the basic random variable that we started with can be constructed as where each edge is working, independently, with probability .
6. The Existence Question
The existence of constant rate distributions for standard posets is an interesting mathematical question. Recall that the poset of finite subsets of under set inclusion, considered in Example 1.9, is associated with a positive semigroup. This positive semigroup does not support exponential distributions , but we have been unable to determine if the poset supports constant rate distributions. We can show that if there is a random subset with constant rate , then the number of elements in must have a Poisson distribution with parameter . A more general question is whether every standard discrete poset without maximal elements supports a constant rate distribution.
- C. M. Goldie and S. Resnick, “Records in a partially ordered set,” Annals of Probability, vol. 17, pp. 678–699, 1989.
- P. R. Halmos, Measure Theory, Springer, Berlin, Germany, 1974.
- T. Kamae and U. Krengel, “Stochastic partial ordering,” Annals of Probability, vol. 6, pp. 1044–1049, 1978.
- L. Nachbin, Topology and Order, Van Nostrand, 1965.
- C. Berge, Principles of Combinatorics, Academic Press, New York, NY, USA, 1971.
- G. Rowell and K. Siegrist, “Relative aging of distributions,” Probability in the Engineering and Informational Sciences, vol. 12, no. 4, pp. 469–478, 1998.
- K. Siegrist, “Exponential distributions on semigroups,” Journal of Theoretical Probability, vol. 7, no. 4, pp. 725–737, 1994.
- K. Siegrist, “Decomposition of exponential distributions on positive semigroups,” Journal of Theoretical Probability, vol. 19, no. 1, pp. 204–220, 2006.
- K. Siegrist, “Random finite subsets with exponential distributions,” Probability in the Engineering and Informational Sciences, vol. 21, no. 1, pp. 117–132, 2007.
- K. Siegrist, “Exponential and gamma distributions on positive semigroups, with applications to Dirichlet distributions,” Bernoulli, vol. 13, no. 2, pp. 330–345, 2007.
- G. Matheron, Random Sets and Integral Geometry, John Wiley & Sons, New York, NY, USA, 1975.
- B. Bollobás, Random Graphs, Cambridge University Press, Cambridge, UK, 2nd edition, 2001.
- M. Drmota, Random Trees, Springer, New York, NY, USA, 2009.
- I. B. Gertxbakh, Statistical Reliability Theory, Marcel Dekker, New York, NY, USA, 1989.
- T. A. Azlarov and N. A. Volodin, Characterization Problems Associated with the Exponential Distribution, Springer, Berlin, Germany, 1986.
- P. S. Puri and H. Rubin, “On a characterization of the family of distributions with constant multivariate failure rates,” Annals of Probability, vol. 2, pp. 738–740, 1974.