Multiplicative Self-Decomposable Laws

• Multiplicative self-decomposable laws are defined by the factorization Z = Z^α Z_α, where the independent residual factor enables a multiplicative analog of additive self-decomposability.
• Mellin-transform techniques are used to derive explicit residual forms for classical families such as exponential, gamma, and half-normal laws.
• Lévy process constructions and exponential functionals provide a complementary framework for constructing and analyzing these positive self-decomposable laws.

Multiplicative self-decomposable laws are positive probability laws that admit factorization into a powered copy of themselves and an independent residual factor. In the explicit formulation now used for positive random variables $Z$, multiplicative self-decomposability means that for every $\alpha\in(0,1)$,

$Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$

where $Z_\alpha$ is independent of $Z$. The same phenomenon can be expressed additively after the logarithmic transform: $\log Z$ is self-decomposable on $\mathbb R$ if and only if $Z$ satisfies such a product decomposition on $(0,\infty)$. Current work develops this subject through two main routes: explicit Mellin-transform factorizations for classical positive laws, and Lévy-process constructions that produce positive self-decomposable laws whose multiplicative interpretation is implicit rather than separately axiomatized (Silva et al., 31 Jul 2025, Patie, 2010, Jedidi et al., 2021).

1. Definition and logarithmic correspondence

The classical additive notion is the class $L_0(\mathbb R)$ of self-decomposable laws, characterized by

$\alpha\in(0,1)$0

with $\alpha\in(0,1)$1 independent of $\alpha\in(0,1)$2. For positive random variables, exponentiation transfers this property to a multiplicative one: $\alpha\in(0,1)$3 Accordingly, the explicit multiplicative definition used in recent work is

$\alpha\in(0,1)$4

with $\alpha\in(0,1)$5 independent of $\alpha\in(0,1)$6. This is the exact parameterization adopted for exponential, gamma, and half-normal laws, and it is also the natural interpretation of earlier results stated only for $\alpha\in(0,1)$7 or for positive self-decomposable laws (Silva et al., 31 Jul 2025, Jedidi et al., 2021).

For positive laws, the natural transform is the Mellin transform

$\alpha\in(0,1)$8

If $\alpha\in(0,1)$9 with independence, then Mellin convolution gives

$Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$0

This ratio formula is the basic analytic mechanism behind explicit residual-law identification. In the same direction, Urbanik’s nested classes $Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$1 induce multiplicative hierarchies on $Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$2: if $Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$3, then $Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$4 inherits an $Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$5-fold multiplicative self-factorization structure, although the papers do not define a separate multiplicative $Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$6-class (Silva et al., 31 Jul 2025, Jedidi et al., 2021).

2. Lévy and exponential-functional constructions

A distinct route to multiplicative self-decomposable laws starts from a possibly killed subordinator $Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$7 with Laplace exponent

$Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$8

and exponential functional

$Z \overset{\mathcal L}= Z^\alpha Z_\alpha,$9

or, in the killed case,

$Z_\alpha$0

The classical Carmona–Petit–Yor moment formula is

$Z_\alpha$1

Bertoin and Yor showed that there exists a positive random variable $Z_\alpha$2 such that

$Z_\alpha$3

and

$Z_\alpha$4

The refined factorization identifies $Z_\alpha$5 as the reciprocal of an exponential functional of a spectrally negative Lévy process. Define

$Z_\alpha$6

If the Lévy measure of the subordinator is absolutely continuous with monotone decreasing density,

$Z_\alpha$7

then $Z_\alpha$8 is the Laplace exponent of a spectrally negative Lévy process with positive mean, and the positive random variable $Z_\alpha$9 determined by

$Z$0

is exactly the associated exponential functional. Consequently $Z$1 is a positive self-decomposable random variable, and

$Z$2

with $Z$3 and $Z$4 independent. The self-decomposability comes from the affine identity

$Z$5

valid for spectrally negative Lévy processes with positive mean. Since $Z$6 is then self-decomposable on $Z$7, $Z$8 provides a canonical positive law with multiplicative self-decomposable interpretation, even though the paper itself does not introduce that terminology explicitly (Patie, 2010).

The same framework links self-similar entrance laws to reciprocals of exponential functionals. If $Z$9 is the time-$\log Z$0 entrance variable of the corresponding self-similar Feller process, then

$\log Z$1

This places reciprocals, quotients, and entrance laws in the same Lévy-exponential-functional architecture (Patie, 2010).

3. Mellin-transform factorization and explicit residual laws

The most explicit multiplicative theory currently available concerns classical one-parameter families and identifies the residual factor $\log Z$2 in closed form. For positive laws, the residual Mellin transform is obtained by division, and in three central cases it can be recognized as the Mellin transform of a named special-function density. This closes the identification gap left by Shanbhag–Sreehari for gamma laws and their descendants (Silva et al., 31 Jul 2025).

Base law	Multiplicative decomposition	Residual law
Exponential $\log Z$3	$\log Z$4	$\log Z$5-Wright
Gamma $\log Z$6	$\log Z$7	Fox $\log Z$8 / Wright
Half-normal $\log Z$9	$\mathbb R$0	Wright

For the exponential law $\mathbb R$1 with density $\mathbb R$2, the residual $\mathbb R$3 has density

$\mathbb R$4

and

$\mathbb R$5

Its Mellin transform is

$\mathbb R$6

while $\mathbb R$7 is Weibull with

$\mathbb R$8

The product of these two transforms is $\mathbb R$9, the Mellin transform of the exponential density (Silva et al., 31 Jul 2025).

For the gamma law

$Z$0

the residual law $Z$1 is defined by the Fox $Z$2-function density

$Z$3

and the multiplicative decomposition is

$Z$4

Its Mellin transform is

$Z$5

The same density admits the Wright-form representation

$Z$6

When $Z$7, the gamma law reduces to the exponential law and the residual Fox $Z$8-density reduces to the $Z$9-Wright density (Silva et al., 31 Jul 2025).

For $(0,\infty)$0, since $(0,\infty)$1, one obtains

$(0,\infty)$2

where

$(0,\infty)$3

These decompositions are not merely existence statements: they characterize the exponential, gamma, and half-normal laws through functional equations for normalized Mellin transforms (Silva et al., 31 Jul 2025).

4. Canonical families and multiple decomposability

The exponential-functional approach yields concrete positive self-decomposable families that are multiplicative in effect. A principal corollary is that if $(0,\infty)$4 is a positive stable random variable of index $(0,\infty)$5, then

$(0,\infty)$6

is a positive self-decomposable random variable. In the same framework,

$(0,\infty)$7

is also a positive self-decomposable random variable, where $(0,\infty)$8 denotes a gamma random variable with parameter $(0,\infty)$9. These results arise from explicit transforms of Lévy exponents,

$L_0(\mathbb R)$0

and from the factorization of the exponential law into independent positive factors (Patie, 2010).

Gamma laws form the main bridge between multiplicative self-decomposability and multiple selfdecomposability. For every $L_0(\mathbb R)$1,

$L_0(\mathbb R)$2

hence

$L_0(\mathbb R)$3

Thus $L_0(\mathbb R)$4 is multiplicatively self-decomposable through the additive self-decomposability of $L_0(\mathbb R)$5. More sharply, $L_0(\mathbb R)$6 is twice selfdecomposable if and only if

$L_0(\mathbb R)$7

This supplies a nontrivial threshold inside the Gamma family for higher-order multiplicative factorization on the log scale (Jedidi et al., 2021).

The same paper derives exact multiplicative factorizations involving weighted geometric products of independent Gamma variables. If $L_0(\mathbb R)$8 with $L_0(\mathbb R)$9, and

$\alpha\in(0,1)$00

then for independent Gamma$\alpha\in(0,1)$01 variables $\alpha\in(0,1)$02: $\alpha\in(0,1)$03 while for $\alpha\in(0,1)$04,

$\alpha\in(0,1)$05

These identities show that multiplicative self-decomposition extends beyond one-factor residual laws to structured geometric products and Gamma-function ratios (Jedidi et al., 2021).

A related consequence is Kanter’s factorization, recovered in this framework as

$\alpha\in(0,1)$06

with additional information that the remainder variable $\alpha\in(0,1)$07 belongs to

$\alpha\in(0,1)$08

This ties positive stable laws, gamma laws, and mixtures of exponentials into the same Mellin-Euler calculus (Jedidi et al., 2021).

5. Adjacent theories and common conflations

The literature contains several neighboring notions that are not equivalent to multiplicative self-decomposability. A central example is multiplicative strong unimodality. For a positive random variable $\alpha\in(0,1)$09, this means that $\alpha\in(0,1)$10 preserves unimodality under independent multiplication by any unimodal factor, and it is equivalent to log-concavity of $\alpha\in(0,1)$11, or equivalently to strong unimodality of $\alpha\in(0,1)$12. For positive $\alpha\in(0,1)$13-stable laws $\alpha\in(0,1)$14,

$\alpha\in(0,1)$15

This is a multiplicative shape property, not a self-factorization property, even though the logarithmic transform again plays the decisive role (Simon, 2010).

Several papers remain strictly additive but are naturally read multiplicatively after exponentiation. For the exponential law, the decomposition

$\alpha\in(0,1)$16

with $\alpha\in(0,1)$17, $\alpha\in(0,1)$18, and all components independent, implies

$\alpha\in(0,1)$19

The original paper uses this to build correlated exponential renewals and correlated Poisson processes, not a theory of multiplicative self-decomposable laws; the multiplicative reading is implicit (Petroni et al., 2015). The same is true for generalized tempered stable laws: the theory is formulated additively through class $\alpha\in(0,1)$20, background driving Lévy processes, and Ornstein–Uhlenbeck stationary laws, while the product decomposition

$\alpha\in(0,1)$21

appears only after setting $\alpha\in(0,1)$22 and exponentiating the additive self-decomposition of $\alpha\in(0,1)$23 (Nzokem, 2024).

Other additive factorization results are structurally suggestive but not multiplicative in the strict sense. Hyperbolic characteristic functions

$\alpha\in(0,1)$24

define selfdecomposable laws and satisfy exact quotient identities such as

$\alpha\in(0,1)$25

but this is additive convolution on $\alpha\in(0,1)$26, not multiplicative convolution on $\alpha\in(0,1)$27 (1009.3542). Likewise, the factorization class $\alpha\in(0,1)$28 is defined by the additive property that a selfdecomposable law convolved with its own background driving law remains selfdecomposable if and only if the background law is $\alpha\in(0,1)$29-selfdecomposable (1009.3545).

A further nonclassical extension appears in free and Boolean probability. There the multiplicative semigroups

$\alpha\in(0,1)$30

on $\alpha\in(0,1)$31, together with their branch-sensitive analogues on $\alpha\in(0,1)$32, define multiplicative free divisibility indicators $\alpha\in(0,1)$33. On the unit circle,

$\alpha\in(0,1)$34

and

$\alpha\in(0,1)$35

These are semigroup-theoretic divisibility thresholds, not classical multiplicative self-decomposability, but they show how decomposition ideas migrate into noncommutative multiplicative convolution theories (Arizmendi et al., 2011).

6. Scope and current limitations

The present theory is rich in construction principles but narrow in classification. The exponential-functional approach requires a specific monotonicity hypothesis: $\alpha\in(0,1)$36 and it does not provide a general characterization of all multiplicatively self-decomposable laws on $\alpha\in(0,1)$37. The corresponding paper explicitly states that it does not define a separate multiplicative self-decomposability notion; the multiplicative reading is inferred from positive self-decomposability and reciprocal or quotient factorizations (Patie, 2010).

The Mellin-transform approach is explicit but currently concentrated on a small set of classical families. Exponential, gamma, and half-normal laws now have identified residual factors in terms of $\alpha\in(0,1)$38-Wright, Wright, and Fox $\alpha\in(0,1)$39-function densities, together with converse characterization theorems, but broader positive classes are not treated in the same closed-form manner (Silva et al., 31 Jul 2025). In the Urbanik-class direction, the theory gives an effective criterion for $\alpha\in(0,1)$40, and hence for iterative multiplicative decomposability of $\alpha\in(0,1)$41, but it stops short of introducing a standalone multiplicative hierarchy on $\alpha\in(0,1)$42 (Jedidi et al., 2021).

The surrounding literature also shows that not every multiplicative regularity concept should be conflated with self-decomposability. Multiplicative strong unimodality, free multiplicative infinite divisibility, and additive factorization properties of class $\alpha\in(0,1)$43 all use products, powers, or logarithms, yet they answer different questions (Simon, 2010, Arizmendi et al., 2011, 1009.3545).

This suggests that the field remains construction-oriented rather than classificatory. Its most solid results currently come from three sources: Lévy exponential functionals and refined factorizations of the exponential law; Mellin-transform identification of residual factors for explicit positive families; and log-side analysis of multiple selfdecomposability through Gamma-function ratios and Urbanik classes (Patie, 2010, Silva et al., 31 Jul 2025, Jedidi et al., 2021).