Abstract

The law of has distribution function and first moment . The law of the length-biased version of has by definition the distribution function . It is known that is infinitely divisible if and only if , where is independent of . Here we assume this relation and ask whether or is infinitely divisible. Examples show that both, neither, or exactly one of the components of the pair can be infinitely divisible. Some general algorithms facilitate exploring the general question. It is shown that length-biasing up to the fourth order preserves infinite divisibility when has a certain compound Poisson law or the Lambert law. It is conjectured for these examples that this extends to all orders of length-biasing.

1. Introduction

Let be a nonnegative random variable whose distribution function (DF) and Laplace-Stieltjes transform (LST) is denoted by and , respectively. If the law of , , is infinitely divisible (infdiv), then , where the Laplace exponent (or cumulant function) is a Bernstein function, denoted by . This means that there is a measure (called the Lévy measure) on satisfying and a constant such that In this case is the left-extremity of the support of and hence we will set , thereby losing no generality. Differentiation of yields which in turn is equivalent to the convolution identity where . Conversely, if a DF satisfies (2) where is a measure having a finite LST, then is the DF of an infdiv law. See Theorem 4.10 in [1] and the reference there to Steutel’s original formulation of this result.

Suppose now that the first moment and let denote the DF of the length-biased (or size-biased) version of . Clearly , and hence is the DF of a random variable, say. So if denotes a random variable having the DF and is infdiv, then (2) has the random variable formulation where denotes equality in law and the random variables on the right-hand side are independent. (Note that it is always assumed that random variables occurring on the right-hand side of in-law equalities are independent.) The equality (3) underlines the fact that length-biasing is an increasing operation with respect to the stochastic order; if . Conversely, if (3) holds for some positive defect random variable , then is infdiv. See [2] for more on this result.

The primary question we address is that if (3) holds, then is also infdiv? A secondary question is whether is infdiv? If it is, then clearly is infdiv. The primary question can be extended to asking whether arbitrary order length-biasing of an infdiv law also is infdiv. Here, if , then we define the order- length-biased version of by , where . Thus and denotes the length-bias operator of order acting on distribution functions having a finite moment of order . We extend this notation by writing for a random variable whose DF is and we denote the corresponding LST by . See [3] for various properties of .

The paper is structured as follows. Definitions of some important classes of infdiv laws are collected in Section 2. Generalities about length-biasing and infinite divisibility are addressed in Section 3. First, several examples show that both, neither, or exactly one member of the pair can be infdiv. Second, if is infdiv, then there is a law such that . If also is infdiv, then so is . This is the approach taken for the particular cases discussed in Sections 4 and 5. Theorem 7 provides a convenient recursive method of computing the LSTs of the defect laws . This section ends with a general additive representation for , assuming that it is infdiv, and also Lemma 8 giving a general formula for the result of applying to the sum of two independent random variables.

In Section 4 we examine the case where has the unit rate compound Poisson law whose jump law is the Exp(35) law and Section 5 deals with the case which first motivated this paper; that is, is the Lambert law introduced by Pakes [4]. As mentioned, the approach in both cases is, having established that is infdiv, to investigate as far as possible the infdiv status of . We thereby establish in both cases that is infdiv for and we conjecture that this is the case for all . Our algebraic manipulations become more involved as increases, suggesting other methods are needed to resolve our conjectures one way or the other.

We end this introduction by reminding the reader that length-biasing is important in contexts such as random sampling of regions in spatial problems, for example, the distribution of the volume or surface area of spheres given the distribution of radii; see page 170 in [5]. Another context is measuring income inequality via the Lorenz curve, that is, the set ; see page 40 in [6]. Many probability laws can be characterized through relations involving length-biasing, as in [3, 7]. The first of this pair of references cites other occurrences of length-biasing.

2. Classes of Infdiv Laws

Recall that a positive-valued function defined and real-analytic on is completely monotone, written , if for all . Bernstein’s theorem asserts that if and only if there is a positive measure such that . In particular, if , then its derivative .

We are principally concerned with infdiv laws where has a density . We say that is self-decomposable, written , if is nonincreasing. Self-decomposable laws comprise the limit laws of affine transformations of sums of independent random variables.

We say that is an exponential mixture law, written , if there is a possibly defective DF on (the mixing DF) such that the corresponding DF . This is equivalent to the random variable equality , where has the standard exponential law Exp(35) and has the DF . Thus, . In addition, is infdiv and its Laplace exponent has the form where .

Following terminology in [8], say that the Laplace exponent is a complete Bernstein function, written , if . The corresponding class of laws are termed Bondesson  (BO) laws in [8] and laws in [9], which term replaces the original descriptive designation  GCMED, meaning “generalised convolution of mixtures of exponential distributions.” This last term underlines the fact that the  BO laws comprise the smallest class which is closed under convolutions and weak limits and contains . Finally, we observe that, like , is closed under composition.

An important subclass of complete Bernstein Lévy exponents is the set of Thorin Bernstein functions defined by the requirement that . Thus, . Note that the composition of two functions in need not be in , although it will be in . Infdiv laws having a Lévy exponent in comprise those laws which are the weak limits of sums of independent gamma distributed random variables, and hence they are called generalized gamma convolutions   (GGCs). The class of  GGCs is the smallest which contains gamma laws and is closed under convolutions and weak limits. They have become important because in recent years the self-decomposability of many familiar continuous laws has been demonstrated by showing that they are  GGCs; see [9] or [1]. The Lévy exponent of a  GGC has the form where the Thorin measure satisfies and .

Computing the derivative of the Laplace exponent (5) and noting that yield the representation Comparing this with the derivative obtained from (4), we obtain the following known result (page 352 in [1]).

Lemma 1. If   GGC, then if and only if .

Finally, we say that a positive-valued function () is hyperbolically completely monotone (HCM) if, for each , the function of equal to is a completely monotone function of . A key result is that a law which has an HCM density is a  GGC and hence is self-decomposable. Example 3 below gives two examples and [9] presents many more. A recent summary with examples of the above ideas is [10].

We end this section with the following result about LSTs having the following form. Let be a discrete law, a sequence of positive numbers, and constants. The gamma mixture LST will arise in examples below. Let have the beta density , where, conventionally, . It is consistent with the moment structure of the beta laws to specify and . Also, let be a random variable having the standard gamma law with density .

Lemma 2. The LST (7) is infdiv with the scale mixture representation where , and and are independent.

Proof. It follows from the well-known gamma-beta identity that Hence the form (7) follows by computing the expectation at (8). The infdiv assertion follows because scale mixtures of the gamma(2) law are infdiv; see page 344 in [1].

We note that scale mixtures of the gamma law may, or may not, be infdiv if ; see page 409 in [1]. Finally, although we will not use this, it is worth observing that a limit argument based on (7) and Lemma 2 shows that if is a DF on and is a function on this set with values in , then is an infdiv LST.

3. General Observations

We begin with examples which illustrate what is possible in relation to the infinite divisibility of and its length-biased versions. The first simply reminds readers that there is a rich collection of infdiv laws whose length-biased versions of all orders also are infdiv.

Example 3. Suppose that has a HCM density . If is a real constant, then also is HCM; see page 68 in [9]. It follows that if is finite, then is infdiv. One example is the generalized inverse Gaussian laws (page 27 in [11]) whose densities , where and . Clearly moments are finite if . The case yields the gamma laws, . In this case we have . If it does not seem possible to identify the law in (3), even in the case .
Suppose that is a real constant, , and has the standard normal law. Then has the lognormal law whose density function . This density function is HCM (page 59 in [9]), and clearly has a lognormal law for all real .
More examples of HCM densities may be found in [9] and the fact that if has a HCM density, then so does if .

The next example shows the existence of laws which are not infdiv for any .

Example 4. It is known that infinite divisibility of a positive law imposes constraints on how fast the survivor function can go to zero as . In fact, if is infdiv, then it follows from (9.6) on page 114 in [1] that for any . Laws with a bounded support comprise the extreme contradiction of this condition; they are not infdiv and neither are their length-biased versions. Laws with unbounded support where where and likewise are not infdiv; their right-hand tail is too thin. Clearly if , and the length-biased laws of all positive orders satisfy a similar tail constraint and hence are not infdiv.

Our next example shows that length-biasing laws which are not infdiv can yield infdiv laws.

Example 5. Let , , and have the LST representing the scale mixture , where has the Bernoulli law with . This LST has the form (7) and hence it is infdiv if , but it is not SD (page 346 in [1]). On the other hand, if then is not infdiv (page 409 in [1]). For any , the order length-biased version of is the gamma law, and hence it is infdiv.
We will look at the case in more detail. The LST of in (3) is Suppose first that and let denote the stable law of index , denoted by stable and having the LST Note that . If is independent of and if is a constant, then the LST of a scaled positive Linnik law (also called the Mittag-Leffler law); see page 219 in [12] (and its references) and page 38 in [9] for a proof that Linnik laws are  GGCs.
Next, let and let denote the exponential tilt operator; . The corresponding LST is , and hence is infdiv if is infdiv. Each of the classes ,  BO, and  GGC is closed under exponential tilting. If has the DF , then denotes a random variable whose DF is .
Still assuming that and referring to (12), it follows that is the LST of , where . We thus can express the defect random variable in the representation (3) as where . Thus   GGC, and if , then we have the simpler representation .
The case is more opaque. In particular, the factor need not be a Laplace transform; it is not convex if . If , some algebra yields where . It follows that has the density It clearly has zeros in and hence is not infdiv [1], although, as remarked above, is the gamma(3) law which is a  GGC.
The next example shows that length biasing an infdiv law can result in a non-infdiv law. This possibility is noted in Example 12.6 of [1] (page 413) for two cases where is not infdiv, although it is not clear whether or not is infdiv. The following example rests on the fact that if is infdiv then if , where is the abscissa of convergence of the integral defining . Note that if can be analytically continued into , then it can have zeros there.

Example 6. If then having the LST is a scale mixture of gamma(2) laws, and hence it is infdiv. This LST is holomorphic in .
Since , it is clear that is a mixture of gamma(3) laws which may, or may not, be infdiv. We show now that the latter can occur. Calculation shows that if , that is, if where, since , The real part of has the same sign as With , the right-hand side bracketed term is the cubic polynomial . We have if and is increasing in . It follows that if . Hence has a conjugate pair of zeros in the half-plane if , and hence is not infdiv even though is infdiv.

Restricting the length-bias order to positive integer values , it is clear that the LST of is . It follows from (3) that is infdiv if and only if there is a defect law such that in which case the LST of is Thus .

The following result provides a recursive computation scheme for which is simpler to use than (22). Assuming the moments are finite, the first equality in (22) defines a real-analytic function irrespective of whether or not is infdiv and, in addition, . The upper bound is a consequence of the stochastic order relation . Thus the function is positive-valued and real-analytic. Of course and if and only if is infdiv.

Theorem 7. (i) If , then
(ii) If also and are infdiv and the latter has a Lévy density and hence a density , then has a density given by

Proof. Since , we have But, . Substituting into the last display and dividing through by yield (23) and Assertion (ii) follows from (i).

The recursion (23) can be used as follows. If is known to be infdiv, then and in principle it can be inverted to yield the density (or DF) of . If also , then is infdiv and hence so is and the process can be continued. However, as shown in Example 5, it is possible that is infdiv even though is not. Although (24) provides in principle a direct evaluation of , deciding whether is infdiv usually requires a consideration of its LST.

We end this section with two simple deductions from (3). First, assuming that is infdiv for , then iterating (21) yields the additive representation Weakening the assumption here to requiring only that is infdiv a second, more complicated representation is implied by the following lemma about the application of to the sum of random variables. We use the following compact notation. Suppose that satisfies for a positive integer and is a random variable taking values in . Then denotes a random variable whose value is with probability .

Lemma 8. Suppose that () are independent random variables having finite moments (), let , and Then is the probability mass function of a random variable, say, and where the random variables on the right-hand side are independent.

Proof. Let denote the LST of and . Applying the Leibniz differentiation rule to this product shows that the LST of is The mixture LST on the right-hand side is that of the right-hand side of (28).

4. A Compound Poisson Law

We illustrate Theorem 7 with an example which also illustrates the mounting algebraic difficulties as is increased. It has a relation to the Lambert example discussed in Section 5. Let be the DF of the unit rate compound Poisson law whose jump law is the Exp(35) law. Thus and the absolutely continuous component of has the density Note that . If then can be normalised to be an honest density. Now , and some algebra shows that

Case . Since is infdiv, (3) holds and differentiation shows that . Thus is the gamma(2) law, and hence itself is infdiv. The Lévy density of the gamma law is , whence the Lévy density of is . This is completely monotone, and hence BO. The second-order derivative of is , which is not completely monotone, so   GGC.

Case . We have from the case that   BO. Since , it follows from Theorem 7 and (23) with that This has the form (7) so Lemma 2 implies that is infdiv. It follows that too is infdiv and also that , where NBern is independent of the gamma random variable. The rational form of yields and hence the Lévy density of is Denoting the unit (Heaviside) step function with jump at by , it is clear that is the Laplace transform of . Hence   BO, but the second-order derivative of changes sign, so   GGC.

Case . It follows from the case that   BO. Using Theorem 7 and (23) with we obtain The partial fraction form inverts to the density

It follows directly from the rational form in (35) that Since the Laplace transform of is , the inverse transform of is Hence , so is a Lévy density and is infdiv. In addition, similar to the above treatment of , has an inverse Laplace transform comprising a linear combination of unit step functions whose right-continuous version is identically zero in and positive-valued in . Hence  , so   BO. The inverse transform of is a linear combination of two positive and two negative point masses, so   GGC.

Case . This case illustrates the growing algebraic complexity of our procedures. It follows from the previous case that   BO. Using (23), (35), and (37), we have The partial fraction form inverts to which we know is a density function because is infdiv.

The rational form of yields Replacing 73 with 72 in the denominator of the subtracted rational term yields a polynomial which factorizes as . It follows that the zeros of are real and that , , and are small positive numbers. In fact , , and . Calculation shows that and hence the inverse Laplace transform of is where

Note first that , , and as . Denote the successive exponential terms (neglecting their signs) in by (). Clearly and a little arithmetic shows that if and only if and its least value is −0.00047 and that if and only if and its least value is −0.00013. Thus we conclude that in and hence that and are infdiv laws.

Finally, the Lévy density has an inverse transform which again is a linear combination of unit step functions which is positive-valued in . Hence   BO, but similar to the above,   GGC. It follows from our calculations that   BO for and we conjecture that this holds for all .

5. The Lambert Law

The principal solution of the functional equation is called the Lambert -function (named after J. H. Lambert (1728–1777) and not the author of [6]). It is a concave-increasing function mapping into with , , and . Its first-order derivative is (from (45)) Many properties and applications are given in [13] and it is classified as an elementary function in [14].

The key property for our purposes is that . See [4, 15, 16] for various proofs of this fact. It is shown in [4] (Theorem 3.1) that where the Lévy density is and is a probability measure with . Consequently, as observed in the remarks following the proof of Theorem 3.1 in [4], .

It follows that there is a probability law   GGC (denoted by in [4]) whose LST is and is named the (standard) Lambert law. In addition, Theorem 3.2 in [4] asserts that . There is a corresponding Lambert subordinator , that is, a Lévy process having nondecreasing sample paths, such that and . Since , the Lambert subordinator is not compound Poisson; its sample paths are nowhere constant step functions.

Lagrange’s reversion of series yields the known expansion converging if and implying that has finite moments of all positive orders, . In particular

Differentiating (49) and using (46) and then (49) we find that the LST of is Knowing that is infdiv, it follows that (3) holds and that the defect LST is The factor is the LST of , the exponentially stopped Lambert subordinator. Here is independent of . It follows that (3) takes the specific form where . The right-hand side is the sum of independent infdiv random variables and hence is infdiv.

This bare conclusion follows from Example in [9] asserting that the length-biased version of a  GGC law is infdiv. The above working yields some extra detail and, in particular, the following refinement.

Theorem 9. If has the Lambert law, then   BO and hence   BO too.

Proof. The Laplace exponent of is . But and because the corresponding Lévy density is . Thus closure of under functional composition implies that , and hence that   BO. Since is the sum of independent  BO random variables and  BO is closed under convolution, the first assertion follows, and the second likewise.

Instead of invoking Theorem 7, computation of for suggests the following result which follows from an easy induction argument from (22).

Lemma 10. The higher-order derivatives of the LST (49) have the form where and the degree- polynomials are determined from the following recurrence: if then

Writing and setting in Lemma 10 yield . Hence (56) yields . Dividing (56) by and letting yield ; that is, . The argument used in part (35) of the proof of Theorem 3.1 in [4] shows that if and . Table 1 lists exact expressions which we will use below.

The quotients defined at (22) have the form and hence the closure of under functional composition yields the following formal assertion.

Theorem 11. Suppose for each that   BO and that is the LST of a law   BO. Then   BO and where and is independent of the Lambert subordinator . A weaker assertion holds where “ BO” is replaced with “is infdiv.”

Theorems 9 and 11 yield a recursive algorithm for computing the rational form of . Checking that in fact it is the LST of a probability law requires resolving it into its partial fraction form. The next theorem gives a first step toward this goal.

Theorem 12. Let and . For , where , , and is a degree- polynomial if , and . In addition , and for ,

Proof. It follows from (56) that implying (60). The numerator in (61) is a polynomial whose degree is no more than . But the definitions of and imply that this numerator vanishes at and hence is a degree- polynomial as asserted. The recursions for and follow from (56).

Table 2 displays values of the above constants for . Asymptotic expressions for and can be derived from (62) and (63) and they confirm the rapid increase suggested by the values in Table 2. The algorithm for is straightforward but time consuming to implement. Table 2 lists formulae for .

We look now at the cases , but before starting, we recall that if is infdiv, then . However, we cannot infer that defined at (58) is completely monotone; this has to be demonstrated.

Case . It follows from Tables 1 and 2 that It follows from Lemma 2 that is infdiv. The rational form yields which inverts to In turn, has the inverse transform . We conclude that   BO and hence from Theorems 9 and 11 that   BO.

Case . Here again an infdiv mixture of gamma laws. Similar to the case in Section 4 we have so defining , we obtain the inverse transform which is clearly positive-valued if . Hence is infdiv and so too is . However it is not clear whether or not , and hence we cannot conclude that   BO.

Case . Proceeding as above using Table 2 yields the form which inverts to

To see that this is a density function, observe first that the trigonometric term has its first zero where ; that is, . Thus in . Critical points of occur where , and the least positive solution is with . It follows that the most extreme negative value of the third term inside the square brackets certainly exceeds Hence is the density of a law .

The rational form of computed using Table 1 yields Laplace transform inverses of the first two terms were computed in the case . Denote the cubic polynomial denominator by . It has a zero at (). It follows that so the partial fraction form of the subtracted rational term and inverting yields where . Write the right-hand side as , where Clearly is bounded above by the sum of the exponential terms, a decreasing function which equals 2.0030 if . Hence , say, for all . Computing actual values of indicates that it is positive and decreasing in ; for example, , , , and .

Thus we can safely conclude that is a Lévy density and hence that is infdiv. It follows that is infdiv for and we conjecture that this holds for all .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.