Gamma–Linnik Convolution

• Gamma–Linnik convolution is the operation of convolving a positive Linnik density with a power kernel, resulting in a Prabhakar Mittag–Leffler kernel with explicit Laplace transforms.
• It bridges probabilistic representations and analytic calculus by uniting stable–gamma mixtures with the analytic structure of Prabhakar functions, key in modeling random tree growth and limit theorems.
• The convolution framework underlies simulation schemes, parameter matching, and extensions to generalized Linnik and Mittag–Leffler laws, providing tools for moment analysis and Markov chain constructions.

Gamma–Linnik convolution is the operation that convolves a positive Linnik density with a power kernel and yields a Prabhakar Mittag–Leffler kernel. In the framework of positive stable random variables and gamma random variables, the positive Linnik law arises from the product $S_{\alpha;z}G_{\gamma,\lambda}^{1/\alpha}$, while the convolution

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$

has the explicit form

$$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$

This identity places the positive Linnik, Prabhakar Mittag–Leffler, and two-parameter Mittag–Leffler families in a single calculational and probabilistic scheme, and it also underlies Lamperti-type laws and the Mittag–Leffler Markov chain used in models of random tree growth (Sibisi, 9 Jul 2025).

1. Stable–gamma setup and the positive Linnik law

The construction begins with a one-sided $\alpha$-stable random variable $S_{\alpha;z}$, $0<\alpha<1$, characterized by

$$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$

together with the scaling relation

$$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$

The gamma law is used in both rate and scale parameterizations. In rate form, $G(\nu,\lambda)$ has Laplace transform

$$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$

The power kernel is written as $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$0 (Sibisi, 9 Jul 2025).

The positive Linnik law is defined by the stable–gamma product

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$1

for independent factors. Its density is the stable–gamma mixture

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$2

and its Laplace transform is

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$3

For $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$4, this becomes $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$5, the generalized positive Linnik law associated with Pakes (Sibisi, 9 Jul 2025).

Object	Construction	Transform
Positive stable $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$6	one-sided $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$7-stable law	$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$8
Gamma $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$9	shape $$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$0, rate $$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$1	$$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$2
Positive Linnik $$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$3	$$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$4	$$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$5
Prabhakar kernel	$$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$6	$$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$7

A useful analytic tool throughout the construction is Pollard’s series for the stable density,

$$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$8

which supports explicit density expansions and mixture identities (Sibisi, 9 Jul 2025).

2. Core convolution identity

The gamma–Linnik convolution is the convolution of the one-sided Linnik density with the power kernel $$\lambda^\gamma x^{\beta-1}E^\gamma_{\alpha,\beta}(-\lambda x^\alpha), \qquad \beta\ge \alpha\gamma>0.$$9. For $\alpha$0,

$\alpha$1

Its Laplace transform is

$\alpha$2

The Prabhakar function $\alpha$3 satisfies

$\alpha$4

and uniqueness of Laplace transforms therefore yields

$\alpha$5

This is the central identity of gamma–Linnik convolution: convolving a positive Linnik law with a power kernel produces the Prabhakar Mittag–Leffler kernel (Sibisi, 9 Jul 2025).

The significance of the identity is structural rather than merely computational. It converts a stable–gamma mixture into an explicit special-function kernel, preserving complete Laplace control. This provides a direct bridge between probabilistic product representations and the analytic calculus of Prabhakar functions. In the terminology of (Sibisi, 9 Jul 2025), mixing and convolution already suffice to generate a rich family of distributions.

3. Mittag–Leffler laws, Lamperti-type laws, and parameter matching

The two-parameter Mittag–Leffler distribution $\alpha$6, $\alpha$7, has density

$\alpha$8

where

$\alpha$9

Its Laplace transform is

$S_{\alpha;z}$0

The connection with gamma–Linnik convolution is obtained by setting

$S_{\alpha;z}$1

so that $S_{\alpha;z}$2. Under this parameter match, the convolution formula reproduces the $S_{\alpha;z}$3 density and Laplace transform directly (Sibisi, 9 Jul 2025).

This route is especially important because it shows that the Mittag–Leffler family is not introduced ad hoc. It emerges from a specific convolutional operation on the positive Linnik law. The same framework yields the identity

$S_{\alpha;z}$4

with independent factors, and the recursion

$S_{\alpha;z}$5

These formulas place beta products and beta mixtures inside the same calculus (Sibisi, 9 Jul 2025).

A four-parameter extension is also available: $S_{\alpha;z}$6 with Laplace transform

$S_{\alpha;z}$7

and moments

$S_{\alpha;z}$8

When $S_{\alpha;z}$9, this reduces to the two-parameter law $0<\alpha<1$0 (Sibisi, 9 Jul 2025).

Lamperti-type laws arise as the $0<\alpha<1$1 specialization of the stable–Mittag–Leffler product $0<\alpha<1$2. For $0<\alpha<1$3, the resulting density is identified with the generalized arcsine family, and the change of variable $0<\alpha<1$4 yields the well-known Beta-type form. In the source exposition, this family is tied to occupation-time limits for Markov processes and is associated with Darling–Kac, Lamperti, Feller, Bertoin, Pitman, and James (Sibisi, 9 Jul 2025).

4. Product representations, simulation, and the Mittag–Leffler Markov chain

Gamma–Linnik convolution is accompanied by several product representations. The positive Linnik law is

$0<\alpha<1$5

while $0<\alpha<1$6 satisfies

$0<\alpha<1$7

For general $0<\alpha<1$8, the density $0<\alpha<1$9 is obtained by polynomial tilting: $$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$0 In the notation recorded in (Sibisi, 9 Jul 2025), the Lamperti-type ratio variable can be written as

$$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$1

The simulation scheme is correspondingly direct. For one-sided stable sampling, Kanter’s algorithm is specified for $$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$2: if $$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$3 and $$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$4, then

$$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$5

and $$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$6. A Linnik sample is then obtained by drawing $$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$7 and $$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$8 independently and returning $$\mathbb{E}\big[e^{-xS_{\alpha;z}}\big]=\exp(-z x^\alpha),\qquad x\ge 0,$$9. For $$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$0, one draws $$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$1 and returns $$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$2 (Sibisi, 9 Jul 2025).

A further structure is the Mittag–Leffler Markov chain. For $$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$3 and $$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$4, the sequence $$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$5 is time-homogeneous with transition density

$$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$6

The kernel is independent of $$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$7 and $$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$8. In (Sibisi, 9 Jul 2025), this property is singled out as key for modeling growth processes in random trees and graphs, including stable trees and preferential attachment graphs, and the same laws are said to appear as $$f_\alpha(u\mid z)=f_\alpha(u z^{-1/\alpha}\mid 1)\,z^{-1/\alpha}.$$9-diversity limits in Poisson–Dirichlet partitions $G(\nu,\lambda)$0 and in occupancy/urn models.

For integer $G(\nu,\lambda)$1, simulation of $G(\nu,\lambda)$2 can proceed by starting from $G(\nu,\lambda)$3 and iterating this time-homogeneous kernel. This suggests a recursive sampling mechanism tailored to growth models where successive increments of $G(\nu,\lambda)$4 represent grafting steps (Sibisi, 9 Jul 2025).

5. Analytic properties

The positive Linnik law $G(\nu,\lambda)$5 belongs to the class of generalized gamma convolutions and is therefore infinitely divisible. Bondesson’s Thorin measure is given with density

$G(\nu,\lambda)$6

In the source exposition, this is derived by Stieltjes inversion of the logarithmic derivative of the Linnik Laplace transform, placing the law inside the Pick-function and GGC framework (Sibisi, 9 Jul 2025).

The Prabhakar kernels generated by gamma–Linnik convolution also have a complete-monotonicity interpretation. For $G(\nu,\lambda)$7, $G(\nu,\lambda)$8, and $G(\nu,\lambda)$9, the functions

$$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$0

appear as Laplace transforms of positive measures, and

$$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$1

is described as a completely monotone kernel in $$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$2 and $$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$3. This is consistent with its use as a probability density or resolvent kernel (Sibisi, 9 Jul 2025).

Moment formulas are explicit. For the two-parameter Mittag–Leffler law,

$$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$4

so all moments are finite. The generalized family $$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$5 has moments

$$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$6

These formulas are obtained by Mellin-transform arguments applied to the Prabhakar Laplace identity (Sibisi, 9 Jul 2025).

Auxiliary asymptotic statements are also recorded. Near the origin,

$$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$7

so the kernels $$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$8 are regular at $$\int_0^\infty e^{-sx}g_{\nu,\lambda}(x)\,dx=\left(\frac{\lambda}{\lambda+s}\right)^\nu,\qquad s\ge 0.$$9 for $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$00. For large $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$01, the source notes that

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$02

uniformly in $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$03 on compact sets, a statement used there in connection with resolvent decay (Sibisi, 9 Jul 2025).

6. Broader usage in generalized Linnik and Mittag–Leffler theory

Related literature uses a broader Gamma–Linnik viewpoint for symmetric generalized Linnik laws. In that setting,

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$04

or, at unit scale,

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$05

defines the generalized Linnik characteristic function. The same transform is represented by gamma–stable mixing: $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$06 with $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$07 symmetric strictly stable. The generalized Mittag–Leffler law on $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$08 is defined in parallel by

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$09

and one of the central mixture identities is

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$10

for $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$11 independent (Korolev et al., 2018).

This viewpoint emphasizes closure under addition of the parameter $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$12. If $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$13 and $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$14 are independent generalized Linnik variables with the same $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$15 and $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$16, then

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$17

so the $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$18-parameters add. In the gamma-mixing representation, this corresponds to the gamma-sum identity

$$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$19

The same papers connect these representations to random-sum limit theorems, asymptotically normal statistics with random sample sizes, maxima of random sums, extreme order statistics, and one-sided or two-sided Mittag–Leffler limits (Korolev et al., 2016).

A plausible implication is that the term “Gamma–Linnik convolution” has acquired two closely related meanings. In the one-sided theory of (Sibisi, 9 Jul 2025), it denotes a literal convolution of a positive Linnik density with a power kernel that produces a Prabhakar Mittag–Leffler kernel. In the generalized symmetric Linnik literature, it denotes gamma mixing through the Laplace or characteristic transform and the resulting $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$20-additive semigroup structure (Korolev et al., 2016, Korolev et al., 2018). Both usages rest on the same stable–gamma mechanism, but they emphasize different operations: additive convolution with $$\{\rho_{\beta-\alpha\gamma}\star \ell_\alpha(\cdot\mid \gamma,\lambda)\}(x)$$21 in the former, and gamma subordination or scale mixing in the latter.