On two extensions of the canonical Feller–Spitzer distribution

We introduce two extensions of the canonical Feller–Spitzer distribution from the class of Bessel densities, which comprise two distinct stochastically decreasing one-parameter families of positive absolutely continuous infinitely divisible distributions with monotone densities, whose upper tails exhibit a power decay. The densities of the members of the first class are expressed in terms of the modified Bessel function of the first kind, whereas the members of the second class have the densities of their Lévy measure given by virtue of the same function. The Laplace transforms for both these families possess closed–form representations in terms of specific hypergeometric functions. We obtain the explicit expressions by virtue of the particular parameter value for the moments of the distributions considered and establish the monotonicity of the mean, variance, skewness and excess kurtosis within the families. We derive numerous properties of members of these classes by employing both new and previously known properties of the special functions involved and determine the variance function for the natural exponential family generated by a member of the second class.


Introduction
The discovery of new infinitely divisible distributions with regularly varying tails is important for the development of distribution theory per se as well as for mathematical modelling, and applications in statistics and to decision theory. Ideally, they should have tractable probability density (or mass) functions (i.e., the p.d.f. and the p.m.f., respectively) and/or Laplace transform. Here, we introduce the following two different stochastically decreasing one-parameter families of positive absolutely continuous infinitely divisible distributions whose upper tails have a power decay.

Vinogradov and Paris Journal of Statistical Distributions and Applications
(2021) 8:3 Page 2 of 25 Definition 1 Given the real-valued parameter ρ > 1/2, consider the following class of positive functions of argument x ∈ R 1 + := (0, +∞): ( 1 ) We define them at the origin by continuity such that It is shown in "Properties of the class X ρ " section that each such function constitutes the p.d.f. of its own non-negative absolutely continuous random variable, which is hereinafter denoted by X ρ and referred to as the generalized ρ-order Feller-Spitzer r.v. of the first type.
Definition 2 Consider a class of positive infinitely divisible distributions which is indexed by the real-valued parameter ρ > 1/2, has the following density of its Lévy measure ν (2) ρ ({·}) on R 1 + : and (104)) at 0 and +∞ yields the fulfillment of (Bertoin (1996), condition (3.2)). Hence, such r.v., which is hereinafter denoted by Y ρ and called the generalized ρ-order Feller-Spitzer r.v. of the second type, is well defined and generates its own subordinator whose corresponding Laplace transform is as follows:

does not have a drift component. Note that a combination of assumption (3) with the well-known asymptotics of the modified Bessel function of the first kind I r (introduced by
For each real ρ > 1/2, the non-negative r.v. Y ρ is absolutely continuous on R 1 + (see Theorem 5.i). Hereinafter, we denote its p.d.f. by p ρ (x).
It is interesting that the Laplace transforms of members of these two new classes of distributions are related by virtue of the integral representation (60), which in turn is closely related to the concept of ρ-function of the generic member of a certain subclass of infinitely divisible distributions considered in (Steutel and van Harn (2004), formula (V.2.3)). (Note that the parameter ρ employed throughout this paper is not related to the letter ρ used in the concept of ρ-function due to Steutel and van Harn (2004).) Both these families which are thoroughly studied in "Properties of the class X ρ " and "Properties of the class Y ρ " sections, respectively, were derived as an outgrowth of the well-known distribution presented in Definition 3 of "Review and some new results for the Feller-Spitzer class of Bessel densities" section, which is hereinafter referred to as the canonical Feller-Spitzer distribution (compare to Examples 1 and 2). This probability law first appeared in (Spitzer (1964), p. 236), but its major properties were presented by Feller (1966a) and Feller (1966b) who termed a wider class the Bessel densities.
It should be pointed out that (Johnson et al. (1994), Section 12.4.4) considered the socalled Bessel function distributions such that the p.d.f. of the generic member of such a class involves an exponentially tilted modified Bessel function of the first type (104) with an odd value of the index (see formula (12.95) therein). However, in contrast to the p.d.f. 's and Lévy densities (i.e., densities of the corresponding Lévy measures) which emerge in Vinogradov and Paris Journal of Statistical Distributions and Applications (2021) 8:3 Page 3 of 25 this work, the power term of the p.d.f. 's from (Johnson et al. (1994), Section 12.4.4) has a positive exponent, whereas it is negative in our case. Hence, the Bessel function distributions considered in (Johnson et al. (1994), Section 12.4.4) should not be confused with ours. See also "Review and some new results for the Feller-Spitzer class of Bessel densities" section for more detail on the terminology. Each member of the two families introduced in Definitions 1-2 can be used as a mixing measure to generalize the corresponding Poisson mixture with a comparable tail behavior of its p.m.f. However, a detailed consideration of their properties is beyond the scope of this paper. See also (Paris and Vinogradov (2020a), formula (1.2)).
In this paper, we will frequently utilize several special functions including the modified Bessel function of the first kind given by (104) as well as the Gauss hypergeometric function and its generalization, which are introduced by Definitions 4 and 5, respectively. For the reader's convenience, numerous definitions and results on special functions employed in this paper are deferred to "Appendix" section. In particular, it contains two new inequalities (108) and (109) for Bessel functions.
This article is not self-contained. Hence, we refer to Letac (1992) and Jørgensen (1997) for a comprehensive description and important examples of natural exponential families (or NEF's), and to Olver et al. (2010) for more detailed information on the relevant special functions. The proof of some subtle analytic results on new properties of the modified Bessel function of the first kind (104) including the inequalities (108) and (109) are given in our work Paris and Vinogradov (2020b).
To conclude the Introduction, we summarize some notation and terminology that will be used in the sequel. First, we follow the custom of formulating various statements of distribution theory in terms of the properties of r.v. 's, even when such results pertain only to their distributions. The acronym "ch.f." is used for a characteristic function. In what follows, the sign " d =" will denote the fact that the distributions of (univariate) r.v. 's coincide, whereas the symbol " d →" will stand for weak convergence. In the sequel, log stands for the natural logarithm of the real or complex argument (In the complex case, log z is understood as its principal value). An empty sum is interpreted as zero.

Review and some new results for the Feller-Spitzer class of Bessel densities
First, we present the following analytic result that can be derived from Feller (1966a).
The above lemma will be used to relate formulas (6), (7) and (8). Separate parts of (5) give an outgrowth to Definitions 1-2 as well as the main concepts of "Properties of the class X ρ " and "Properties of the class Y ρ " sections.
It is known that the identity (5) is closely related to an important positive absolutely continuous infinitely divisible distribution which we call the canonical Feller-Spitzer distribution, since it was first considered by (Spitzer (1964), p. 236), Feller (1966a), and Feller (1966b). It is introduced as follows: By Feller (1966a) and Feller (1966b), r.v. U 1 is infinitely divisible having the following density τ 1 (x) of its Lévy measure (or its Lévy density): In view of (5), the Laplace transform of U 1 acquires the following form for λ ≥ 0: The variance function V 1 (μ) of the NEF constructed starting from the canonical Feller-Spitzer r.v. U 1 with p.d.f. f 1 (x) is as follows: The validity of (9) was stated by (Letac (1987), p. 154) in his discussion of (Jørgensen (1987), Example 2.1). This also follows from a combination of (Jørgensen (1997), formula (2.18)) with the fact that the inverse of the mean-value mapping for a NEF generated by r.v. U 1 equals We refer to (Jørgensen (1997), p. 48) for more detail on the variance function of a generic NEF and a related concept of the mean-value mapping. The motivation behind the consideration of r.v. U 1 in (Spitzer (1964), p. 236) and (Feller (1966a), Subsection II.7.b) was partly because it emerges as the law of the first passage time of level 1 for the so-called continuous-time symmetric Bernoulli random walk which is characterized by mean 1 exponential time in between the jumps. More specifically, if {B , ≥ 1} are i.i.d.r.v. 's such that P{B = ±1} = 1/2 independent of a Poisson process {N (t), t ≥ 0} with unit intensity, consider the (continuous-time and discrete space) compound Poisson Lévy process S(t) d = B 1 + ... + B N (t) and the corresponding first passage time of positive integer level n ≥ 1: Then the r.v. T n has the p.d.f. f * n 1 (u) of the n th partial sum The p.d.f. f * t 1 (u) was found by (Spitzer (1964), p. 236), Feller (1966a);Feller (1966b) to be as follows: (It is straightforward to check that the expression which emerges on the right-hand side of (11) constitutes a legitimate p.d.f. on R 1 + for an arbitrary real-valued t ∈ R 1 + .) The cumulant-generating function (or the c.g.f.) of the corresponding compound Poisson-Bernoulli r.v. S(1) equals Subsequently, in view of a slight modification of (Jørgensen (1997), pp. 124-125, Exercise 3.15.4), the variance function of the NEF constructed starting from the r.v. with c.g.f. (12) equals Remark 1 It is straightforward to verify that the pair of the variance functions given by formulas (13) and (9) constitutes the reciprocal pair in the sense of (Letac and Mora (1990), Theorem 5.2.iii). In turn, this is consistent with a comment concerning the fluctuation properties of the right-continuous random walk in Spitzer's sense, which was made in the paragraph that precedes (Letac and Mora (1990), Theorem 5.6). See also (Spitzer (1964), Definition 1.2.3) for the definition of the latter random walk as well as a comment that follows this definition.
Note in passing that Feller considered a more general class of p.d.f. 's on R 1 + , some of which became important in applications. For instance, it contains the distribution of the busy period in M/M/1 queue (compare to (Stewart (2009), p. 530)).

Properties of the class X ρ
First, recall that this class was introduced in Definition 1 of "Introduction" section.
The following assertion stipulates that the family {X ρ , ρ > 1/2} possesses an important property of stochastic monotonicity. The interested reader is referred to Cohn (1981) and Lee et al. (2009) for a connection between this concept and various convergence properties, and for relevant statistical tests and applications, respectively (see also the references therein). Its relationship to weak convergence is illustrated just above Propositions 3 and 7. Also, note in passing that in the context of a family of positive r.v. 's, stochastic monotonicity is equivalent to monotonicity of the corresponding family of the survival functions at each value of the argument.

Theorem 1
The family {X ρ , ρ > 1/2} of the generalized Feller-Spitzer distributions of the first type is stochastically decreasing in the sense that for an arbitrary fixed real y > 0, the function ρ → P{X ρ > y} is decreasing on (1/2, ∞).
Proof of Theorem 1. In view of the above comment, it suffices to prove monotonicity of the family of the values of the survival functions for each value of argument y ∈ R 1 + . A subsequent combination of formula (15) with Kummer's transformation (see (Olver et al. (2010), formula (13.2.39)) yields that for arbitrary fixed ρ > 1/2 and y ∈[ 0, +∞), the survival function F ρ (y) of the r.v. X ρ admits the following representation: where t n (ρ) : Finally, as the sum in F ρ (y) consists of positive terms, and the coefficients t n (ρ) are positive decreasing functions of ρ, it follows that the survival function F ρ (y) must decrease with increasing ρ.
Proof of Proposition 1. First, the following integral representation for the function 2 F 1 can be derived with some effort from (Watson (1952), Section 13.2, p. 385, formula (2)) by implementing the change of variables b → ib in this formula and using the fact that the Bessel function of the first kind J r (ibt) = e πri/2 I r (bt): A subsequent combination of (18) with the values of a = 1, b = 1/(λ + 1), c = ρ + 1, k = 1 and (1)-(2) implies the validity of (17). Compare to the derivation of (Paris and Vinogradov (2020a), formula (5.13)) where the case of c = 2 was considered.
Evidently, the closed-form expression which emerges in (17) generalizes formula (8) which pertains to the case where ρ = 1.
Example 1 For ρ = 1, formula (1) turns into (6). Hence, the distributions of r.v. X 1 and the canonical Feller-Spitzer r.v. U 1 coincide, i.e., It is clear that in the case where ρ = 2, the p.d.f. of the r.v. X 2 is such that for x ∈ R 1 + , Also, a combination of (17) with (Paris and Vinogradov (2020a), formula (4.1)) yields that for a fixed λ ∈[ 0, +∞), It is relevant that by analogy with (21), it is possible to derive the closed-form representations for L X ρ (λ) for numerous other values of ρ > 1/2 some of which follow from (Paris and Vinogradov (2020a), formula (4.1)).

Theorem 2 Fix an arbitrary real
and (v) The following recursive formula holds for x ∈ R 1 + : Proof of Theorem 2. (i) It follows from the fact that for ρ > 1/2, the distribution of the r.v. X ρ satisfies the log-convexity property, which easily follows from inequality (108) of Theorem 8.i (see Corollary 1 of "Appendix" section for more detail).
(iii) The proof of (24) follows from a combination of (2) and (22). (iv) Let us replace ρ by ρ + 1 in (1) and combine the corresponding representation for f ρ+1 (x) with (Paris and Vinogradov (2020b), formula (4.4) and the unnumbered formula just below (4.6)). We obtain that In view of (Paris and Vinogradov (2020b), formula (4.5)), the expression which emerges on the right-hand side of the above equation equals The result then follows upon insertion of the definition (1) of f ρ (x).
(ii) The r.v. V admits the following compound geometric representation: independent of a geometric counting r.v. Geom(2/3) (that takes values in the set of all nonnegative integers), which is characterized by its Laplace transform , where λ > − log 3.
Hereinafter, the p.d.f. f * 2 1 (u) which emerges in the middle of formula (30) denotes the two-fold convolution of the p.d.f. f 1 (u) of the r.v. U 1 (compare to (11)).
Proof of Proposition 2. First, one should verify the following identity: where the Laplace transforms which emerge in identity (32) are given by formulas (26), (31) and (29). This verification is straightforward and hence left to the reader. In turn, (32) implies the validity of representation (28). The rest follows from a rather obvious folk theorem that a compound geometric sum of i.i.d.r.v. 's is infinitely divisible (compare to (Kyprianou (2010), claim above formula (8)). Although X ρ is infinitely divisible ∀ρ > 1/2, it is quite challenging to find its Lévy representation for ρ = 1. Recall that for ρ = 1, it is given by (8).
Hereinafter, γ denotes the Euler-Mascheroni constant (see (106)). Note in passing that in view of (34) and as y → +∞, which is consistent with (33) and (Embrechts et al. (1979), Theorem 1). We believe that representation (34) for the Lévy measure ν (1) 1 ({(y, ∞)}) in terms of the function 3 F 3 was previously unknown. See also Proposition 6, which extends formulas (34)-(35) for all the ρ-order Feller-Spitzer r.v. 's of the second type introduced in Definition 2. Also, both a closed-form representation for the cumulative distribution function of r.v. U 1 in terms of the confluent hypergeometric function 1 F 1 and the asymptotics of its upper tail can be derived by setting ρ = 1 in formulas (15) and (33), respectively.
Next, Theorem 1 implies with some effort existence of the weak limit for the stochastically decreasing family X ρ as ρ → +∞, which should be a continuous distribution on R 1 + . In turn, part (i) of the following assertion identifies this limit as the mean 1 exponential r.v., which is hereinafter denoted by E, whereas part (ii) can be regarded as a local limit theorem on the exponential convergence.

Vinogradov and Paris Journal of Statistical Distributions and Applications
(2021) 8:3 Page 10 of 25 Next, it easily follows from a combination of (6) with formula (107) of Lemma 3 that given real s ∈ (−1, ρ − 1/2), the s th raw moment of the r.v. X ρ is as follows: Subsequently, we employ the above formula (39) to provide the closed-form representations for the mean, variance, skewness and excess kurtosis for those members of the class {X ρ , ρ > 1/2} for which the corresponding numerical characteristic(s) are finite.
(ii) The second, third and fourth central moments of r.v. X ρ admit the following closedform representations.
An application of the representations of Proposition 4.ii with the help of Mathematica stipulates that the skewness (1) 1 (ρ) and the excess kurtosis (1) 2 (ρ) of the r.v. X ρ are as follows. For ρ > 7/2, the skewness decreases from +∞ to 2 as ρ increases from 7/2 to +∞. Also, for ρ > 9/2, the excess kurtosis decreases from +∞ to 6 as ρ increases from 9/2 to +∞. (40), (41), (44) and (45) stipulate that as ρ → +∞, the mean, variance, skewness and excess kurtosis of the respective members of the class {X ρ } converge to 1, 1, 2 and 6, respectively, which are the values of the corresponding numerical characteristics of the mean 1 exponential r.v. E. This observation on the moment convergence is consistent with the integral and local theorems on the exponential convergence presented in Proposition 3 above.

Properties of the class Y ρ
First, recall that this class was introduced in Definition 2 of "Introduction" section. This second extension of the canonical Feller-Spitzer distribution U 1 (introduced in Definition 3) concerns its Lévy density τ 1 (x) given by (7), which is in contrast to the first extension {f ρ (x), ρ > 1/2} which pertains to the p.d.f. f 1 (x) per se.
Next, a combination of (3) with (Olver et al. (2010), formula (10.30.4))) implies that the Lévy density τ ρ (x) (which is introduced by formula (3)) exhibits the following power decay as x → +∞: (compare to (14) and (71)). On the other hand, it follows with some effort from a combination of (3) with (Olver et al. (2010), formulas (10.25.2) and (10.30.1))) that this function diverges at the origin exhibiting the following asymptotic behavior at the right neighborhood of zero: The following result is similar in spirit to Theorem 1 of "Properties of the class X ρ " section.
Theorem 3 The family {Y ρ , ρ > 1/2} of the generalized Feller-Spitzer distributions of the second type is stochastically decreasing in the sense that for an arbitrary fixed real y > 0, the function ρ → P{Y ρ > y} is decreasing on (1/2, ∞).
In order to prove Theorem 3, we first present the following general and straightforward but rather technical assertion which might already be known.

Vinogradov and Paris Journal of Statistical Distributions and Applications
(2021) 8:3 Page 12 of 25 Lemma 2 Suppose that for i = 1, 2 there exist two positive functions τ (i) (·) on R 1 + such that the following integrals are well-defined and finite: Assume that for i = 1, 2 there exist two non-negative infinitely divisible r.v. 's Y i with zero drifts and diffusion components which possess the Lévy densities on R 1 + given by functions τ (i) (·), respectively, such that for real λ ≥ 0, Then (i) There exists a non-negative infinitely divisible r.v. K independent of Y 2 such that (ii) For an arbitrary fixed y ∈ R 1 + , Proof of Lemma 2.
(i) It follows from the assumptions of the lemma and (49) that for real λ ≥ 0, In view of (48)-(49), the expression which emerges on the left-hand side of (52) constitutes the negative of the log-Laplace transform of a certain non-negative infinitely divisible r.v. denoted by K whose density of Lévy measure equals τ (1) (x) − τ (2) (x), which implies (50).
(ii) The validity of (51) easily follows from (50). Proof of Theorem 3. First, let us prove that for a fixed x ∈ R 1 + , the Lévy density τ ρ (x) defined by (3) is a decreasing function of the parameter ρ on (1/2, +∞). To this end, we rewrite this function as follows: Hence, the above expression is obviously monotonically decreasing in ρ for a fixed real x > 0 as each term after the first is a decreasing function of ρ. The rest easily follows Lemma 2.ii.

Remark 3
Fix the values of 1/2 < ρ 1 < ρ 2 < +∞. Then a combination of formulas (46)-(47) yields that Hence, in the case where the r.v. Y i which emerges in Lemma 2 coincides with the generalized ρ-order Feller-Spitzer r.v. of the second type Y ρ i (for i = 1, 2), the non-negative Vinogradov and Paris Journal of Statistical Distributions and Applications (2021) 8:3 Page 13 of 25 infinitely divisible r.v. K (= K ρ 1 ,ρ 2 ) which is present on the right-hand side of representation (50) is in fact compound Poisson. For a representative special case for which ρ 1 = 1 and ρ 2 = 2, the corresponding compound Poisson r.v. K (= K 1,2 ) is identified in Remark 5 of "A case study: properties of the r.v. Y 2 " section.
(ii) For an arbitrary fixed ρ > 1/2 and as y → +∞, Proof of Proposition 6. (i) The validity of (54) can be established by the use of Mathematica. The analytic proof is given in (Paris and Vinogradov (2020b), Lemma 1).
(ii) Observe that the leading term of the asymptotic series which emerges on the righthand side of (53) is as follows: Next, combine formulas (3), (46), (56), the leftmost equation in (54), and the well-known fact that the power asymptotics of the function τ ρ (x) at positive infinity implies the corresponding power asymptotics of its integrated tail. This yields the validity of the rightmost asymptotics in expression (55). The rest follows from an application of (Embrechts et al. (1979), Theorem 1).
The following assertion is of particular value.
The following example is in a similar spirit to that of Example 1.

Example 2 For ρ = 1, a comparison of formula (3) of Definition 2 with representations (6)-(8) of Definition 3 stipulates that in this case,
Hence, the distributions of r.v. 's Y 1 and the canonical Feller-Spitzer r.v. U 1 coincide, i.e., (compare to formula (19)). Next, note that the digamma function ψ is defined by formula (105). It is well known that ψ(1) = −γ and ψ(1/2) = −γ − 2 · log 2. A combination of these two representations with (63) and (Prudnikov et al. (1990), Section 7.4.2, entry 365) to evaluate the 3 F 2 function stipulates that Evidently, (66) coincides with (58) in the case where ρ = 1. It is interesting that although the middle expression which emerges in formula (60) is not defined for ρ ∈ (1/2, 3/2], but in the case where ρ = 1 the equality of the left-and right-hand sides on that formula can be derived analytically. In this case, (compare to the right-hand side of (63)).
A detailed consideration of the case where ρ = 2 is presented separately in "A case study: properties of the r.v. Y 2 " section. In particular, see Theorem 7 and Remark 6 therein which demonstrate how complex the p.d.f. 's of the r.v. 's Y ρ can be for ρ = 1.
The following result is of a particular value providing an illustration of the usefulness of numerous relatively recent advances in the theory of infinitely divisible distributions, most of which were summarized in (Steutel and van Harn (2004), Chapter V). To some extent, it can be regarded as a counterpart of Theorem 2 of "Properties of the class X ρ " section.
Theorem 5 Fix an arbitrary ρ > 1/2. Then (i) The non-negative r.v. Y ρ is self-decomposable and absolutely continuous on R 1 + . (ii) The limit from the right at the origin for the p.d.f. p ρ (x) of the r.v. Y ρ is as follows: The rest coincides with the end of the proof of Proposition 3.i. Let us introduce the following quantities: The following assertion provides the counterparts of the closed-form representations (40)-(43) (which pertain to the numerical characteristics of r.v. X ρ ).
(ii) The second, third and fourth central moments of r.v. Y ρ admit the following closed-form representations.
An application of the representations of Proposition 8 with the help of Mathematica ascertains that the skewness (2) 1 (ρ) and the excess kurtosis (2) 2 (ρ) of the r.v. Y ρ are as follows. For ρ > 7/2, the skewness decreases from +∞ to 2 as ρ increases from 7/2 to +∞. In addition, for ρ > 9/2, the excess kurtosis decreases from +∞ to 6 as ρ increases from 9/2 to +∞. Remark 4 (i) Similar to the large-ρ behavior of the numerical characteristics of the class {X ρ } pointed out in Remark 2 of "Properties of the class X ρ " section, representations (81), (82), (84) and (85) stipulate that as ρ → +∞, the mean, variance, skewness and excess kurtosis of the respective members of the class {Y ρ } converge to 1, 1, 2 and 6, respectively, which are the values of the corresponding numerical characteristics of the mean 1 exponential r.v. E. Evidently, this observation is consistent with the weak convergence result of Proposition 7.
The following assertion provides a closed-form expression for the Lévy representation of the r.v. Y 2 in terms of elementary functions and connects its p.d.f. p 2 (x) with the members of the class (11).
Proof of Proposition 9. Combine (4), (58) and (88). In turn, the above representation for 2 (λ) has a probabilistic interpretation (90) which specifies decomposition (50) in the case where Y i d = Y i (for i = 1, 2) and is described in the following remark, which can be viewed as a counterpart to Proposition 2.

Remark 5
The expression 1 2 · (e −2 1 (λ) − 1) which emerges in (89) is recognized as the log-Laplace transform of a particular non-negative compound Poisson r.v. By analogy to representation (50), it is hereinafter denoted by K. Specifically, the r.v. K admits the following (compound Poisson) representation: