 Research
 Open Access
On Poisson–Tweedie mixtures
 Vladimir V. Vinogradov^{1}Email authorView ORCID ID profile and
 Richard B. Paris^{2}
https://doi.org/10.1186/s4048801700681
© The Author(s) 2017
 Received: 27 February 2017
 Accepted: 11 July 2017
 Published: 2 October 2017
Abstract
PoissonTweedie mixtures are the Poisson mixtures for which the mixing measure is generated by those members of the family of Tweedie distributions whose support is nonnegative. This class of nonnegative integervalued distributions is comprised of Neyman type A, backshifted negative binomial, compound Poissonnegative binomial, discrete stable and exponentially tilted discrete stable laws. For a specific value of the “power” parameter associated with the corresponding Tweedie distributions, such mixtures comprise an additive exponential dispersion model. We derive closedform expressions for the related variance functions in terms of the exponential tilting invariants and particular special functions. We compare specific PoissonTweedie models with the corresponding HindeDemétrio exponential dispersion models which possess a comparable unit variance function. We construct numerous local approximations for specific subclasses of PoissonTweedie mixtures and identify Lévy measure for all the members of this threeparameter family.
Keywords
 Discrete stable distribution
 Invariant of the exponential tilting transformation
 Lambert W function
 Large deviations
 Lévy measure
 Natural exponential family
 PoissonTweedie mixture
 Refined local limit theorem
 Variance function
 Wright function
Mathematics Subject Classification (2010)
 33E20
 60E05
 60E07
 60F05
 60F10
Introduction
In this paper, we establish new results of distribution theory and prove new limit theorems of probability theory. Specifically, we investigate and establish numerous properties of the threeparameter family of nonnegative integervalued random variables (or r.v.’s) which are hereinafter referred to as the PoissonTweedie mixtures. This family was considered, among many others, by Kokonendji et al. (2004, Section 3), Jørgensen and Kokonendji (2016), and Bonat et al. (2017). The PoissonTweedie mixtures are rigorously introduced by formula (3).
We concentrate on the derivation of local limit theorems, which is customary in the case where one deals with integervalued r.v.’s, since in view of the jumps of their cumulative distribution functions, the integral limit theorems for such r.v.’s are usually less accurate, which is due to discontinuities to be taken care of. Moreover, local limit theorems often provide a more detailed picture of the convergence mechanism than their integral counterparts by pointing out at potential singularities. For instance, Remark 4 to Theorems 7 and 8 addresses the singularity at the origin for the local versions of the (integral) theorem on weak convergence for PoissonTweedie mixtures given by formula (50) – the fact that can only be revealed through the microscope of local behavior.
Members of the PoissonTweedie family defined by formula (3) are often used for modeling overdispersed count data, since the variance of a generic member of this class is greater than its mean (compare to formula (4)). At the same time, the simulation studies presented in Bonat et al. (2017, Section 4) pertain to fitting extended PoissonTweedie regression models to overdispersed and underdispersed data. In turn, a growing interest to this family within the statistics community along with a close connection of the probability function of a general member of this class to the Wright special function (presented by formula (30)) motivated us to consider subtle mathematical properties of the PoissonTweedie mixtures. Thus, we concentrate on the theoretical aspects rather than applications of the PoissonTweedie mixtures and rely on the abovequoted papers for the raison d’être. In many cases, the relationship (30) between PoissonTweedie mixtures and Wright function (6) makes it possible to derive the leading error term of our local approximations whose role in applications is yet to be determined.
Next, similar to Kokonendji et al. (2004) we denote this class of PoissonTweedie mixtures by \(\{{\mathcal {P}}{\mathcal {T}}_{p, \mu, \lambda }\}\), although a different notation, namely, \(\{{\mathcal {P}}{\mathcal {T}}_{p}(\theta, \lambda)\}\), is employed by Kokonendji et al. (2004, Section 3). This is because we construct the class of PoissonTweedie mixtures by virtue of formula (3) starting from the corresponding members of the reproductive Tweedie exponential dispersion models (or EDM’s) for which the variancetomean relationship is given by formula (2). In contrast, Kokonendji et al. (2004, formula (9)) derived the probability law of a generic member \({\mathcal {P}}{\mathcal {T}}_{p}(\theta, \lambda)\) of the PoissonTweedie class starting from the corresponding representative of a specific additive Tweedie EDM. See Jørgensen (1997, Chapters 3 and 4) respectively, for discussion on these forms of EDM’s in the general setting and in the context of Tweedie EDM’s. Our notation is more convenient for the derivation of limit theorems.
See Jørgensen (1997, Chapters 2 and 4) for more details on NEF’s and Tweedie distributions, respectively. (In this paper, we should exclude the case of negative values of the power parameter which correspond to Tweedie laws taking values in the entire real axis R ^{1}, since it is customary to employ nonnegative probability laws only for constructing Poisson mixtures, compare to formula (3).) Also, the variancetomean relationship (2) justifies referring to the totality of Tweedie distributions as the powervariance family or the PVF (compare to Vinogradov et al. (2012, 2013)).
A combination of (2)–(3) yields that \(\mathbf {E}({\mathcal {P}}{\mathcal {T}}_{p, \mu, \lambda }) = \mu \).
Since by (4) the variance is greater than the mean, all the PoissonTweedie mixtures are overdispersed. Also, note that Kokonendji et al. (2004, Proposition 2) states that the increasing function Φ _{ p }(μ) is “generally implicit” being the inverse of function K ^{′}(μ) which they defined by formula (10) therein.
In contrast, our Theorem 1 provides several closedform expressions which specify the variancetomean relationship for all the members of the PoissonTweedie family \(\left \{{\mathcal {P}}{\mathcal {T}}_{p, \mu, \lambda }, p \geq 1, \mu \in \Omega _{p}, \lambda \in \mathbf {R}_{+}^{1} \right \}\) introduced by formula (3). The representations given in that theorem involve the invariants (20) of the exponential tilting transformation and particular special functions. This approach employs the fact that indexing a specific variance function by invariant(s) of the exponential tilting transformation for a fixed p provides a convenient decomposition of the corresponding twoparameter class of the PoissonTweedie distributions into the union of nonoverlapping NEF’s, with each specific NEF corresponding to its own value of the invariant. We defer the consideration of a few special cases and a detailed comparison of Theorem 1 with some related work to Section 4. For instance, Remark 5 addresses a comparison our closedform representations (37)–(39) with “generally implicit” formula (4) and some other related results.
Although the expressions (37)–(39) are interesting in their own right, they can also be used for the derivation of the exact asymptotics of the probabilities of large deviations of partial sums of PoissonTweedie r.v.’s in the case where the magnitude of these deviations is at least proportional to the growing number of the summands (compare to Paris and Vinogradov (2015, Corollary 3.8)). See, for example, representation (40) of Theorem 2, which can be regarded as a result of the saddlepoint approximation type. The subsequent local limit Theorems 3 and 4 which pertain to the values of p∈(1,2) and p>2, respectively, present the exact asymptotics of superlarge deviations for the corresponding partial sums of lattice r.v.’s. Theorems 2 and 3 can be regarded as the results of Cramér’s type, whereas the mechanisms of formation of the probabilities of large deviations in the cases covered by Theorem 4 are qualitatively different. Specifically, representation (42), which pertains to the lattice distributions, is of the same character as numerous results on large deviations for nonlattice r.v.’s presented in Vinogradov (1994, Chapter 5), and (2008b, Theorem 3.6.ii).
Theorems 5 and 6 concern local asymptotics in the case where the corresponding classes of PoissonTweedie mixtures converge to a Poisson limit. In this respect, observe that Kokonendji et al. (2004, Table 2) suggests that as p→+∞, members of a certain subclass of the family of PoissonTweedie mixtures tend to a Poisson law. This is clarified by formula (48), which easily follows from Proposition 1. A local version of this assertion is presented as Theorem 6. See also Remark 3.ii and Conjecture 1. In particular, formulas (82) and (84), which were checked numerically, specify the leading error term of the local Poisson approximation applicable in the case of sufficiently large values of the “power” parameter p.
Proposition 3 addresses the behavior of the PoissonTweedie mixtures around the points p=1 and 2, whereas Theorems 7 and 8 provide local approximations in the case where the PoissonTweedie mixtures \(\{{\mathcal {P}}{\mathcal {T}}_{p, \cdot, \cdot }\}\) converge to a Tweedie distribution with the same p. Since all the PoissonTweedie mixtures are infinitely divisible, the abovedescribed limit theorems of Section 3 can be regarded as the results on local asymptotics for the marginals of specific exponential families of (compound Poisson) integervalued Lévy processes.
Propositions 1 and 2 of Section 2 provide the probabilitygenerating function (or the p.g.f.) of all the PoissonTweedie mixtures and their Lévy measure, respectively.
In Section 4, we compare the PoissonTweedie family with a different class of the additive HindeDemétrio EDM’s which correspond to a simpler u.v.f. given by (61). All the proofs are deferred to “Appendix 1” section, whereas “Appendix 2” section presents two relevant conjectures, which are of independent interest.
Notation, definitions and basic properties
First, we summarize some notation and terminology that will be used in the sequel. We follow a 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. In what follows, the symbol “\(\stackrel {d}{\rightarrow }\)” stands for weak convergence, whereas log denotes the natural logarithm of the real argument. Also, N and \(\mathbb {C}\) stand for the sets of all positive integers and the complex plane, respectively.
We now introduce several special functions and polynomials.
Definition 1
Hereinafter, we refer to ϕ as the (complexvalued) “reduced” Wright function.
Definition 2
where real k≥0. The function _{1} Ψ _{1} constitutes a particular case of the (complexvalued) Wright function.
Under the restriction δ=0, the Wright function (6) admits a representation in terms of the complete Bell polynomials, which is stipulated by Proposition 5. We introduce them as follows:
Definition 3
(see, for example,Paris (2016)). It is well known that the Touchard polynomials can be expressed in terms of the complete Bell polynomials such that for 1≤j≤k, the argument z _{ j }=x: T _{ k }(x)≡B _{ k }(x,x,…,x).
Definition 4

(i) The complexvalued Lambert function W(z) is defined as the multivalued inverse of the function y(x):=x·e ^{ x }. Equivalently, it can be defined as the function satisfying the identity W(z)·e ^{ W(z)}≡z, where \(z \in \mathbb {C}\). Its Taylor series around z=0,$$ W(z) = \sum_{\ell = 1}^{\infty} w_{\ell} \cdot z^{\ell}, $$(9)
has the radius of convergence 1/e. The coefficients w _{ ℓ } of the Taylor series (9) are as follows: w _{ ℓ }=(−ℓ)^{ ℓ−1}/ℓ!.

(ii) The series (9) can be extended to a holomorphic function on \(\mathbb {C}\) with a branch cut along (−∞,−1/e]. This function defines the principal branch W _{ p }(z) of W(z).
It can be shown that for a fixed p≥1, the quantity \({\mathcal Z}_{p}\) is an invariant of the exponential tilting transformation for the corresponding class of PoissonTweedie mixtures with such p. This means that given p≥1, all the members of the class of PoissonTweedie distributions characterized by the same value of \({\mathcal Z}\) comprise their own natural exponential family (or NEF). See Jørgensen (1997, Chapter 2) for more details on NEF’s.
Here, the function \(u^{1} \cdot \phi (\rho, 0, {\mathcal C} \cdot u^{\rho })\) is extended at zero as u ↓0 by continuity, where \({\mathcal C} \in \mathbf {R}_{+}^{1}\) is a constant. For p>2, the law of the r.v. Tw _{ p }(μ,λ) is obtained from that of stable r.v. Tw _{ p }(∞,λ) by the exponential tilting transformation.
Remark 1
Definition 5
Proposition 1
 (i)For p∈(1,+∞)∖{2}, \(\mu \in \mathbf {R}_{+}^{1}\), \(\lambda \in \mathbf {R}_{+}^{1}\), u<θ _{ p }+1 if p∈(1,2), and u≤θ _{ p }+1 if p>2,$$ {\mathcal{P}}(u) = \exp\left\{{\mathcal A}_{p} \left(\left(1 + \frac{1u}{\theta_{p}}\right)^{\rho_{p}}  1 \right)\right\} = e^{{\mathcal Z}_{p, \infty} \left\{ (1 + \theta_{p} u)^{\rho_{p}}  \theta_{p}^{\rho_{p}}\right\} }. $$(32)
 (ii)For p>2, μ=+∞, \(\lambda \in \mathbf {R}_{+}^{1}\), and u≤1,$$ {\mathcal{P}}(u) = \exp\left\{ {\mathcal Z}_{p, \infty} (1u)^{\rho_{p}}\right\}. $$(33)
 (iii)In the case where p=1, \(\mu \in \mathbf {R}_{+}^{1}\), \(\lambda \in \mathbf {R}_{+}^{1}\), and for u∈R ^{1},$$ {\mathcal{P}}(u) = \exp \left\{{\mathcal A}_{1} \cdot \left\{ e^{\lambda^{1} (u 1)}  1 \right\} \right\}. $$(34)
 (iv)For p=2, \(\mu \in \mathbf {R}_{+}^{1}\), \(\lambda \in \mathbf {R}_{+}^{1}\), and u<θ _{2}+1,$$ {\mathcal{P}}(u) = (1 + 1/\theta_{2}  u/\theta_{2})^{\lambda}. $$(35)
Proposition 2
Main results
The first result of this section concerns the closedform representations for the variance functions of specific NEF’s comprised of particular PoissonTweedie mixtures. Note that by (20), \({\mathcal Z}_{p} > 0\) if p∈[1,2], whereas \({\mathcal Z}_{p} < 0\) if p>2.
Theorem 1
 (i)In the case where 1<p<2 and \({\mathcal Z}_{p} > 0\), the variance function of such NEF is as follows:$$ \mathbf{V}_{{\mathcal Z}_{p}}(\mu) = \mu + \frac{\mu^{2}}{(2p) {\mathcal Z}_{p} \cdot t_{s0}(\mu/(\rho_{p} {\mathcal Z}_{p}))^{\rho_{p}}}, $$(37)
where the argument \(\mu \in \mathbf {R}^{1}_{+}\) and \(t_{s0}(\mu /(\rho _{p} {\mathcal Z}_{p}))\) is obtained from (10)–(11) by setting r=ρ _{ p } and \(w = \mu /(\rho _{p} {\mathcal Z}_{p})\).
 (ii)Given p>2 and \({\mathcal Z}_{p} < 0\), the variance function of such NEF admits the following representation:$$ \mathbf{V}_{{\mathcal Z}_{p}}(\mu) = \mu + \frac{\mu^{2}}{(2p) {\mathcal Z}_{p}} \cdot y_{s}\left(\rho_{p} {\mathcal Z}_{p}/\mu\right)^{\rho_{p}}, $$(38)
where the argument μ∈(0,+∞] and \(y_{s}(\rho _{p} {\mathcal Z}_{p}/\mu)\) is derived from (14) by setting ρ=ρ _{ p } and \(a = \rho _{p} {\mathcal Z}_{p}/\mu \).
 (iii)For p=1 and \({\mathcal Z}_{1} > 0\), the variance function of such NEF comprised of specific Neyman type A distributions is as follows:$$ \mathbf{V}_{{\mathcal Z}_{1}}(\mu) = \mu \cdot (1 + W_{p}(\mu/{\mathcal Z}_{1})), \; {\text{where}} \; \mu \in \mathbf{R}^{1}_{+}. $$(39)
Next, we proceed with three local large deviation limit theorems for n ^{ th } partial sums of the i.i.d.r.v.’s whose common distribution belongs to the PoissonTweedie family. The first of them employs the above variance function \(\mathbf {V}_{{\mathcal Z}_{p}}(\cdot)\) given by (37).
Theorem 2
Theorem 3
Theorem 4
The following assertion generalizes Paris and Vinogradov (2015, Theorem 3.13) and refines the local counterpart of the Poisson convergence result (43).
Theorem 5
The following result is related to Kokonendji et al. (2004, Table 2).
Theorem 6
Remark 3
It is easily seen that under the fulfillment of (45), the condition that \(\mu _{p} > \lvert {\mathcal Z}\rvert + \epsilon \) for all sufficiently large p is sufficient for (46), whereas the condition \(\mu _{p} > \lvert {\mathcal Z}\rvert  \epsilon \) is necessary. (Here, ε>0 is an arbitrary small real.)
(ii) In the case of discrete stable distributions per se, i.e., when μ=μ _{ p }≡+∞ and condition (46) is fulfilled automatically, it is plausible to derive the leading error term in the local Poisson convergence result (47). By (33), the discrete stable distributions with the same fixed value of \({\mathcal Z}_{p, \infty } (= {\mathcal Z}) < 0\) converge weakly as ρ _{ p } ↓−1 to a Poisson distribution with mean \(\lvert {\mathcal Z}_{p, \infty }\rvert \). (By convention, Poisson r.v.’s are often included to the class of discrete stable r.v.’s.) See also Conjecture 1.
The next assertion can be regarded in some sense as a local counterpart of (49).
Proposition 3
The case p=1 is to be treated separately, since by (39), \(\mathbf {V}_{{\mathcal Z}_{1}}(\mu) \sim \mu \cdot \log \,\mu \) as μ→+∞ rather than just to Const·μ (compare to formula (62)).
The following assertion, which can be regarded as a refinement of the local version of (50) in the case where p=2, constitutes a result of the Yaglomtheorem type on gamma convergence (compare to Jørgensen et al. (2009, pp. 411412)).
Theorem 7
Theorem 8
Remark 4
one ascertains that in the case where p>2 (or ρ _{ p }∈(−1,0)), the expression (53) approaches 0 faster than any negative power of \({\mathcal C}\). Hence, taking limit as \({\mathcal C} \rightarrow +\infty \) eliminates the point mass at the origin in this case.
Since the behavior of the gamma density f _{2,μ,λ }(u) at zero is similar to that of f _{ p,μ,λ }(u) and determined by a particular value of the shape parameter λ, we elected to exclude the value of argument u=0 from consideration in Theorem 7 and give it as a separate formula (54).
(ii) There exists a function \(\{{\mathcal E}_{p, \mu, \lambda }(u),~u > 0\}\) which constitutes the next term in the expansion (52) over negative powers of \({\mathcal C}\). Similar to function \({\mathcal D}_{p, \mu, \lambda }(u)\), it admits a representation in terms of the “reduced” Wright function which is analogous to, but more complicated than, the expression that emerges on the righthand side of formula (51). Hence, it is too long to be included here. Compare to Paris and Vinogradov (2015, Theorem 3.10) where this function is given in the special case where p=3/2.
(iii) The firstorder error terms which emerge in Theorems 7 and 8 are consistent in the sense that \({\mathcal D}_{2, \mu, \lambda }(u) = {\lim }_{p \rightarrow 2} {\mathcal D}_{p, \mu, \lambda }(u)\).
Special cases of Theorem 1, HindeDemétrio EDM’s and discussion
First, we will present a few special cases of Theorem 1 as well as discuss its relationship to Kokonendji et al. (2004) and other works in a series of remarks.
Remark 5
A subsequent combination of representation (20) for \({\mathcal Z}_{1}\) with Jørgensen (1997, Subsection 3.3.3) yields that representation (55) is consistent with (39). Also, representation (39) is consistent with Vinogradov (2013, Theorem 5.1).
(ii) In the case where p=3/2, Paris and Vinogradov (2016, formula (2.1)) yields that formula (30) coincides with Paris and Vinogradov (2015, formulas (3.2)–(3.3)). For p=3/2, Paris and Vinogradov (2015, formula (3.15)) implies that in this special case, our representation (37) can be simplified as follows: \(V_{{\mathcal Z}_{3/2}} (\mu) = \mu \cdot \sqrt {4 {\mathcal Z}_{3/2}^{1} \cdot \mu + 1}\). For \({\mathcal Z}_{3/2} = 4\), this formula is consistent with Jørgensen (1997, p. 170).
In the case where \({\mathcal Z}_{3} = \sqrt {2}\), representation (56) is consistent with Kokonendji and Khoudar (2004, formula (3.10)).
Remark 6
(which is consistent with formulas (21), (37), (39) and the L ^{2} analogue of (49)) is equivalent to Conjecture 2.
A combination of (60) with Jørgensen (1997, Theorem 4.5) justifies the validity of (50).
Also, a combination of formulas (12), (15) and (37)–(39) implies that given p∈[1,+∞), \(\mathbf {V}_{{\mathcal Z}_{p}}(\mu) \sim \mu \) as μ ↓0, which is consistent with (43) (see also Jørgensen (1997, Proposition 4.12 and the last formula of Section 4.6)).
(see, for example, Kokonendji et al. (2004, Theorem 5)). As per the followup paper by Kokonendji et al. (2007, p. 278), “the origin of the HindeDemétrio family could be considered as an approximation (in terms of the unit variance function) to the PoissonTweedie family.” In this respect, we should point out that in the case where p=1, even the similarity between their u.v.f.’s does not hold, which in turn necessitates a modification of Kokonendji et al. (2004, Proposition 6.ii) in this particular case. See Remark 7 and formula (62) specifically for more details.
Remark 7
This corrects Kokonendji et al. (2004, Proposition 2.ii) in the case where p=1. Thus, Kokonendji et al. (2004, Proposition 2.ii) holds for p>1 only, in which case it is consistent with formula (60). Moreover, although it is not stated in Kokonendji et al. (2004, Proposition 2), but for a fixed p>1 and as μ ↓0, the u.v.f.’s \(\mathbf {V}^{{\mathcal {P}}{\mathcal {T}}}_{p}(\mu)\) and \(\mathbf {V}^{{\mathcal H}{\mathcal D}}_{p}(\mu)\) are also equivalent to each other, since they are both locally Poisson at zero.
Proposition 4
A comparison of the fractions which emerge in formulas (64) and (65) stipulates that the power decay of the probability function of r.v. \({\mathcal S}{\mathcal A}_{a,\infty }\) as n→+∞ is of order −3/2, but with different factors of proportionality for even and odd n.
In contrast to (64)–(65), in the case where μ = +∞ formula (77) of the next “Appendix 1” section implies that for the corresponding Poissoninverse Gaussian subclass of the PoissonTweedie family, for which the power decay of the probability function at +∞ is also of order −3/2, the factor of proportionality is identical for both even and odd terms. Apparently, the latter class of the Poissoninverse Gaussian laws would hence be a more preferred choice for fitting the data than the exponentially tilted strictly arcsine distributions which can be generated from (63).
However, we reckon that the probability function of a general member of the HindeDemétrio class (with p≠2) still deserves being studied, since this might potentially reveal even more surprising properties which could be of interest for probability theory. (It is well known that the classes \({\mathcal {P}}{\mathcal {T}}_{2}\) and \({\mathcal H}{\mathcal D}_{2}\) coincide being comprised of negative binomial laws, whereas for other values of p, even the ranges of the corresponding subclasses of PoissonTweedie and HindeDemétrio families are different). But a comprehensive comparison of these two classes is beyond the scope of this paper.
Appendix 1. Proofs and auxilliary analysis results
Proof of Proposition 1
is obtained by combining formula (3) with Jørgensen (1997, formula (4.16)) and Panjer and Willmot (1992, formula (8.2.3)). □
Proof of Proposition 4
as real z→+∞ (cf. e.g., Askey and Roy (2010, formulas (5.11.12)–(5.11.13))).
Proposition 5
Proof of Proposition 5
Proof of Proposition 2
It easily follows by rewriting the exponent from the middle expression in (32) with subsequent expansion of the function \((1  u/(\theta _{p}+1))^{\rho _{p}}\phantom {\dot {i}\!}\) in the Taylor series around 0 which can be easily derived from (68). Note that since the signs of \({\mathcal Z}_{p}\) and (ρ _{ p })_{ k } (which emerge in (36)) coincide, the Lévy measure ν _{ p,μ,λ }({k})>0 for each k∈N. □
Proposition 6
 (i)In the case where \(z \in \mathbb {C}\) is fixed and as ρ→+∞,$$ \rho^{\ell} \cdot e^{z} \cdot {~}_{1}\Psi_{1}(\rho, \ell; \rho, 0; z) = \rho^{\ell} \cdot \mathbf{B}_{\ell}(z \cdot (\rho)_{1}, z \cdot (\rho)_{2},\ldots, z \cdot (\rho)_{\ell}) \rightarrow T_{\ell}(z). $$(69)
 (ii)Fix \({\mathcal C} \ne 0\), and assume that ρ→0. Then$$ {~}_{1}\Psi_{1}(\rho, \ell; \rho, 0; {\mathcal C}/\rho) \sim e^{{\mathcal C}/\rho} \cdot ({\mathcal C})_{\ell}. $$(70)
Proof of Proposition 6
(i) By (66), for a fixed n∈N and as ρ→+∞, the ratio \(\Gamma (\rho n+k)/\Gamma (\rho n) = (\rho n)^{k}\{1 + {\mathcal O}(1/\rho)\}\). Hence, the sum appearing in the function _{1} Ψ _{1}(ρ,k;ρ,0;z) reduces in this limit to \(\sum _{n=1}^{\infty } n^{k} z^{n}/n! = e^{z} T_{k}(z)\).
by application of Bressoud (2010, formula (26.8.7)). A subsequent combination of (73) with (71) implies that given k∈N with ρ→0 and \(z \sim {\mathcal C}/\rho \), (70) holds. □
Proof of Proposition 3
It easily follows from a combination of formulas (26)–(27) and (29)–(30) for the probability function of the corresponding PoissonTweedie mixtures with Proposition 6. In particular, the expressions which emerge on the lefthand sides of formulas (69) and (70) are closely related to the probability function of the Poisson–Tweedie mixtures with p∈(1,2)∪(2,+∞), whereas their limits are employed in formulas (26)–(27) which pertain to the cases where p=1 and 2, respectively. □
In the remaining cases, one should solve Eq. (20) for λ given the fixed set of values of p, μ and \({\mathcal Z}_{p}\). In the case where p=1, the solution is found by following along the same lines as the proof of Vinogradov (2013, Theorem 5.1).
Subsequently, the cases where 1<p<2 and p>2 are reduced to solving the Eqs. (10) and (13), respectively, and involve an application of the corresponding analytic results established in Paris and Vinogradov (2016). The details are left to the reader. □
Lemma 1
satisfying all the conditions (i)–(v) imposed in Definition 5.
Furthermore, it can be shown that g ^{‴}(x _{0})=0 and g ^{ iv }(x _{0})<0, where \(x_{0}^{\nu }=3(\rho +1)^{2}/((2\rho +3){\mathcal {A}}_{p})\). It is easily verified that x _{0}>x _{1}>x _{2}>x _{3}.
Proof of Theorem 2 First, Lemma 1 justifies the applicability of Nagaev (1998, Theorem 2) on the exact asymptotics of the probabilities of large deviations. The rest follows from a combination of representation (37) for the variance function \(\mathbf {V}_{{\mathcal Z}_{p}}(\cdot)\) with Jørgensen (1997, p. 50 and Exercise 2.25). □
Proof of Theorem 3 The proof of the fact that the asymptotics is given by the expression which emerges on the righthand side of (41) then easily follows from a combination of (31) with Paris and Vinogradov (2016, formulas (4.8)–(4.9)). □
Lemma 2
Proof of Lemma 2 It involves a combination of representation (30) with Paris and Vinogradov (2016, formula (4.14)). In addition, the proof can be derived from Christoph and Schreiber (1998, formula (9)). Moreover, an application of Paris and Vinogradov (2016, formula (4.13)) or Christoph and Schreiber (1998, formula (9)) makes it possible to construct asymptotic expansions of the probability function that emerges on the lefthand side of formula (77). □
Proof of Theorem 4 For μ=∞, (42) follows with some effort from a combination of Borovkov and Borovkov (2008, p. 167 and Theorem 3.7.1) with (31) and (77). For μ<∞, (42) is easily reduced to the “boundary” case of μ=∞. □
Proof of Theorem 7 The first step involves a combination of formulas (25) and (28) with subsequent derivation of an integral representation for the probability of interest, where the integral over \(\mathbf {R}_{+}^{1}\) is similar to those considered in Vinogradov (2008a, formulas (27) and (32)) and Paris and Vinogradov (2015, formula (3.31)).
The second step relies on an application of Paris (2011, p. 14, formula (1.2.22)) for the derivation of an asymptotic expansion of such an integral. This is carried out by a refinement of the Laplace method and is parallel to Paris and Vinogradov (2015, Proof of Theorem 3.10). The details are left to the reader. □
Next, combine the Poissonmixture representation (28) with Paris and Vinogradov (2016, formula (1.5)). The asymptotics of the integral over \(\mathbf {R}_{+}^{1}\) which will emerge as a result of an application of the latter structural relationship given by Paris and Vinogradov (2016, formula (1.5)) is evaluated by following along the same lines as Paris and Vinogradov (2015, Proof of Theorem 3.10), where the special case of p=3/2 was considered. Again, it relies on a refinement of the Laplace method described in Paris (2011, p. 14, formula (1.2.22)). The details are left to the reader. □
Appendix 2. Relevant conjectures and their numerical verification
In this section, we will present two conjectures pertaining to the behavior of the Wright function _{1} Ψ _{1}(ρ,ℓ,ρ,0;·) and “reduced” Wright function ϕ(ρ,0;·) in the cases where the parameter ρ approaches −1 and +∞, respectively.
Clearly, \(\lambda _{p} \sim e^{1} \cdot (p1) \cdot \lvert {\mathcal Z}\rvert ^{p1}\) as p→+∞.
The following conjecture can be regarded as a prospective refinement of the local limit Theorem 6 on Poisson convergence.
Conjecture 1
The veracity of (85) was checked numerically with the help of Mathematica. For simplicity, set \({\mathcal Z} =  3\) and ℓ=5 (with ε:=1+ρ approaching 0).
The accuracy of the “leading error term” approximation for _{1} Ψ _{1}(ρ,5,ρ,0;−3) as ρ ↓−1
ε=1+ρ  Lefthand side of (85) 

10^{−1}  3.254222 
10^{−2}  3.601987 
10^{−3}  3.637964 
10^{−4}  3.641574 
10^{−5}  3.641935 
10^{−6}  3.641971 
The next (previously unknown) hypothesis relates the “reduced” Wright function ϕ(ρ,0;·) with ρ→+∞ to the principal branch W _{ p } of the Lambert W function.
Conjecture 2
where functions ϕ and W _{ p } are introduced in Definitions 1 and 4, respectively.
The computations summarized in Table 2 suggest that the error (88) decreases to 0 as ρ increases to +∞.
We now provide the “probabilistic” arguments in support of (86). First, it can be shown that Conjecture 2 is equivalent to (59). Then in view of continuity of the PoissonTweedie family in Lévy metric with respect to p (see (49)), it is plausible that their variance functions also converge pointwise as p ↓1. Recall that in the cases where p∈(1,2) and p=1, parts (i) and (iii) of Theorem 1 yield that they are expressed in terms of functions ϕ and W _{ p }, respectively. A subsequent combination of representations (37) and (39) with the anticipated pointwise convergence of the variance functions as p ↓1, (21) and some algebra suggests the validity of (59).
Declarations
Acknowledgements
We thank Martin Mohlenkamp for his advice on TEXing. VVV is grateful to the Fields Institute, University of Toronto and York University for their hospitality during his work on the paper. He also acknowledges the financial support from the Ohio University International Travel Fund for a visit to the second author in the UK. We are thankful to two anonymous referees for a very constructive feedback and numerous suggestions which eventualy led to the inclusion of formulas (2)–(4), Proposition 2, Theorem 2, revision of Theorem 6, Section 4, and Conjecture 1.
Funding
Supported by the Ohio University International Travel Fund.
Authors’ contributions
VVV dealt with the overall presentation, proofs of the probability theory results, and drafted the manuscript. RBP dealt with the asymptotics, proofs of the analysis results, and the numerical verification of two conjectures. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Askey, RA, Roy, A: Gamma function. In: Olver, FWJ, Lozier, DW, Boisvert, RF, Clark, CW (eds.) NIST Handbook of Mathematical Functions, Ch. 5, pp. 135–47. Cambridge University Press, Cambridge (2010).Google Scholar
 Bonat, WH, Jørgensen, B, Kokonendji, CC, Hinde, J, Demétrio, CGB: Extended Poisson–Tweedie: properties and regression models for count data. Stat. Modelling (2017). in press.Google Scholar
 Borovkov, AA, Borovkov, KA: Asymptotic Analysis of Random Walks: HeavyTailed Distributions. Cambridge University Press, Cambridge (2008).View ArticleMATHGoogle Scholar
 Bressoud, DM: Combinatorial analysis. In: Olver, FWJ, Lozier, DW, Boisvert, RF, Clark, CW (eds.)NIST Handbook of Mathematical Functions, Ch. 26, pp. 617–36. Cambridge University Press, Cambridge (2010).Google Scholar
 Christoph, G, Schreiber, K: Discrete stable random variables. Statist. Probab. Lett. 37, 243–7 (1998).MathSciNetView ArticleMATHGoogle Scholar
 Corless, RM, Gonnet, GH, Hare, DEG, Jeffrey, DJ, Knuth, DE: On the Lambert W function. Adv. Comput. Math. 5, 329–59 (1996).MathSciNetView ArticleMATHGoogle Scholar
 Jørgensen, B: The Theory of Dispersion Models. Chapman & Hall, London (1997).MATHGoogle Scholar
 Jørgensen, B, Kokonendji, CC: Discrete dispersion models and their Tweedie asymptotics. AStA Adv. Stat. Anal. 100, 43–78 (2016).MathSciNetView ArticleGoogle Scholar
 Jørgensen, B, Martínez, JR, Vinogradov, V: Domains of attraction to Tweedie distributions. Lith. Math. J. 49, 399–425 (2009).MathSciNetView ArticleMATHGoogle Scholar
 Kokonendji, CC, DossouGbété, S, Demétrio, CGB: Some discrete exponential dispersion models: PoissonTweedie and HindeDemétrio classes. SORT: Stat Oper. Res. Tansactions. 28, 201–14 (2004).MATHGoogle Scholar
 Kokonendji, CC, Demétrio, CGB, Zocchi, SS: On HindeDemétrio regression models for overdispersed count data. Stat. Methodol. 4, 277–91 (2007).MathSciNetView ArticleMATHGoogle Scholar
 Kokonendji, CC, Khoudar, M: On strict arcsine distribution. Commun. Statist. Theor. Meth. 33, 993–1006 (2004).MathSciNetView ArticleMATHGoogle Scholar
 Letac, G, Mora, M: Natural real exponential families with cubic variance functions. Ann. Stat. 18, 1–37 (1990).MathSciNetView ArticleMATHGoogle Scholar
 Nagaev, AV: Large deviations for sums of lattice random variables under the Cramer conditions. Discrete Math. Appl. 8, 403–19 (1998).MathSciNetView ArticleMATHGoogle Scholar
 Panjer, HH, Willmot, GE: Insurance Risk Models. Society of Actuaries, Schaumburg (1992).Google Scholar
 Paris, RB: Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents. Cambridge University Press, Cambridge (2011).View ArticleMATHGoogle Scholar
 Paris, RB: The asymptotics of the Touchard polynomials. Math. Aeterna. 6, 765–79 (2016).Google Scholar
 Paris, RB, Vinogradov, V: Branching particle systems and compound Poisson processes related to PólyaAeppli distributions. Commun. Stoch. Anal. 9, 43–67 (2015).MathSciNetGoogle Scholar
 Paris, RB, Vinogradov, V: Asymptotic and structural properties of special cases of the Wright function arising in probability theory. Lithuanian Math. J. 56, 377–409 (2016).MathSciNetView ArticleMATHGoogle Scholar
 Vinogradov, V: Refined Large Deviation Limit Theorems. Longman, Harlow (1994).MATHGoogle Scholar
 Vinogradov, V: On the powervariance family of probability distributions. Commun. Statist. Theor. Meth. 33, 1007–29 (2004). Errata p. 2573.MathSciNetView ArticleMATHGoogle Scholar
 Vinogradov, V: On infinitely divisible exponential dispersion model related to Poissonexponential distribution. Commun. Statist. Theor. Meth. 36, 253–63 (2007).MathSciNetView ArticleMATHGoogle Scholar
 Vinogradov, V: On approximations for two classes of Poisson mixtures. Statist. Probab. Lett. 78, 358–66 (2008a).MathSciNetView ArticleMATHGoogle Scholar
 Vinogradov, V: Properties of certain Lévy and geometric Lévy processes. Commun. Stoch. Anal. 2, 193–208 (2008b).MathSciNetMATHGoogle Scholar
 Vinogradov, V: Some utilizations of Lambert W function in distribution theory. Commun. Statist. Theor. Meth. 42, 2025–43 (2013).MathSciNetView ArticleMATHGoogle Scholar
 Vinogradov, V, Paris, RB, Yanushkevichiene, O: New properties and representations for members of the powervariance family. I. Lithuanian Math. J. 52, 444–61 (2012).MathSciNetView ArticleMATHGoogle Scholar
 Vinogradov, V, Paris, RB, Yanushkevichiene, O: New properties and representations for members of the powervariance family. II. Lithuanian Math. J. 53, 103–20 (2013). Erratum, ibid. 54, 229 (2014).MathSciNetView ArticleMATHGoogle Scholar