--- title: Multiplicative Self-Decomposable Laws url: https://www.emergentmind.com/topics/multiplicative-self-decomposable-laws type: topic --- # Multiplicative 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 [2507.23467], [1005.4011], [2103.10160]. ## 1. Definition and logarithmic correspondence The classical additive notion is the class \(L_0(\mathbb R)\) of self-decomposable laws, characterized by \[ X \stackrel d= cX + V_c,\qquad c\in(0,1), \] with \(V_c\) independent of \(X\). For positive random variables, exponentiation transfers this property to a multiplicative one: \[ \log Z \stackrel d= \alpha \log Z + Y_\alpha \quad\Longleftrightarrow\quad Z \overset{\mathcal L}= Z^\alpha e^{Y_\alpha}. \] Accordingly, the explicit multiplicative definition used in recent work is \[ Z \overset{\mathcal L}= Z^\alpha Z_\alpha,\qquad \alpha\in(0,1), \] with \(Z_\alpha\) independent of \(Z\). 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 \(\log Z\) or for positive self-decomposable laws [2507.23467], [2103.10160]. For positive laws, the natural transform is the Mellin transform \[ (\mathcal M f)(z)=\int_0^\infty f(x)x^{z-1}\,dx. \] If \(X\overset d=X^\alpha R_\alpha\) with independence, then Mellin convolution gives \[ \mathcal M \rho_{R_\alpha}(z) = \frac{\mathcal M \rho_X(z)} {\mathcal M \rho_X(\alpha(z-1)+1)}. \] This ratio formula is the basic analytic mechanism behind explicit residual-law identification. In the same direction, Urbanik’s nested classes \(L_n(\mathbb R)\) induce multiplicative hierarchies on \((0,\infty)\): if \(\log X\in L_n(\mathbb R)\), then \(X\) inherits an \(n\)-fold multiplicative self-factorization structure, although the papers do not define a separate multiplicative \(L_n\)-class [2507.23467], [2103.10160]. ## 2. Lévy and exponential-functional constructions A distinct route to multiplicative self-decomposable laws starts from a possibly killed subordinator \(\xi\) with Laplace exponent \[ \phi(u)=bu+\int_0^\infty (1-e^{-ur})\,\nu(dr)+q,\qquad u\ge 0, \] and exponential functional \[ I_\phi=\int_0^\infty e^{-\xi_s}\,ds, \] or, in the killed case, \[ I_\phi=\int_0^{\mathbf e_q} e^{-\xi_s}\,ds. \] The classical Carmona–Petit–Yor moment formula is \[ \mathbb E[I_\phi^n] = \frac{\Gamma(n+1)}{\prod_{k=1}^n \phi(k)}, \qquad n=1,2,\dots. \] Bertoin and Yor showed that there exists a positive random variable \(J\) such that \[ \mathbb E[J^n]=\prod_{k=1}^n\phi(k),\qquad n\ge 1, \] and \[ I_\phi J \stackrel{(d)}{=} \mathbf e. \] The refined factorization identifies \(J\) as the reciprocal of an exponential functional of a spectrally negative Lévy process. Define \[ \psi_1(u)=u\phi(u+1). \] If the Lévy measure of the subordinator is absolutely continuous with monotone decreasing density, \[ \nu(dx)=f(x)\,dx, \] then \(\psi_1\) is the Laplace exponent of a spectrally negative Lévy process with positive mean, and the positive random variable \(I_{\psi_1}\) determined by \[ \mathbb E[I_{\psi_1}^{-n}]=\prod_{k=1}^n \phi(k),\qquad n=1,2,\dots, \] is exactly the associated exponential functional. Consequently \(I_{\psi_1}\) is a positive self-decomposable random variable, and \[ \frac{I_\phi}{I_{\psi_1}} \stackrel{(d)}{=} \mathbf e, \] with \(I_\phi\) and \(I_{\psi_1}\) independent. The self-decomposability comes from the affine identity \[ I_\psi \stackrel{(d)}{=} \int_0^{T_y} e^{-\Xi_s}\,ds + e^{-y} I_\psi', \] valid for spectrally negative Lévy processes with positive mean. Since \(\log I_{\psi_1}\) is then self-decomposable on \(\mathbb R\), \(I_{\psi_1}\) provides a canonical positive law with multiplicative self-decomposable interpretation, even though the paper itself does not introduce that terminology explicitly [1005.4011]. The same framework links self-similar entrance laws to reciprocals of exponential functionals. If \(J_\psi\) is the time-\(1\) entrance variable of the corresponding self-similar Feller process, then \[ J_\psi \stackrel{(d)}{=} \frac{1}{I_{\psi_2}}, \qquad \psi_2(u)=\frac{u}{u+1}\psi(u+1). \] This places reciprocals, quotients, and entrance laws in the same Lévy-exponential-functional architecture [1005.4011]. ## 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 \(R_\alpha\) 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 [2507.23467]. | Base law | Multiplicative decomposition | Residual law | |---|---|---| | Exponential \(Y_0\) | \(Y_0 \overset{\mathcal L}= Y_0^\beta Y_\beta\) | \(M\)-Wright | | Gamma \(Z_r\) | \(Z_r \overset{\mathcal L}= Z_r^\alpha Y_{\alpha,r}\) | Fox \(H\) / Wright | | Half-normal \(|U|\) | \(|U| \overset{\mathcal L}= |U|^\alpha X_\alpha\) | Wright | For the exponential law \(Y_0\) with density \(e^{-t}\mathbf 1_{t>0}\), the residual \(Y_\beta\) has density \[ M_\beta(t)=\sum_{n=0}^\infty \frac{(-t)^n}{n!\,\Gamma(-\beta n+1-\beta)},\qquad t\ge 0, \] and \[ Y_0 \overset{\mathcal L}= Y_0^\beta\, Y_\beta,\qquad 0<\beta<1. \] Its Mellin transform is \[ (\mathcal M M_\beta)(z)=\frac{\Gamma(z)}{\Gamma(\beta(z-1)+1)}, \] while \(Y_0^\beta\) is Weibull with \[ (\mathcal M \rho_{Y_0^\beta})(z)=\Gamma(\beta(z-1)+1). \] The product of these two transforms is \(\Gamma(z)\), the Mellin transform of the exponential density [2507.23467]. For the gamma law \[ \rho_{Z_r}(t)=\frac1{\Gamma(r)}t^{r-1}e^{-t},\qquad t>0, \] the residual law \(Y_{\alpha,r}\) is defined by the Fox \(H\)-function density \[ \rho_{Y_{\alpha,r}}(t) = H_{1,1}^{1,0}\!\left( t\,\middle|\, \genfrac{}{}{0pt}{}{(r-\alpha,\alpha)}{(r-1,1)} \right),\qquad t>0, \] and the multiplicative decomposition is \[ Z_r \overset{\mathcal L}= Z_r^\alpha\,Y_{\alpha,r},\qquad \alpha\in(0,1). \] Its Mellin transform is \[ \mathcal M \rho_{Y_{\alpha,r}}(z) = \frac{\Gamma(r-1+z)}{\Gamma(r-\alpha+\alpha z)}, \qquad \Re(z)>1-r. \] The same density admits the Wright-form representation \[ \rho_{Y_{\alpha,r}}(t)=t^{r-1}W_{-\alpha,r(1-\alpha)}(-t),\qquad t>0. \] When \(r=1\), the gamma law reduces to the exponential law and the residual Fox \(H\)-density reduces to the \(M\)-Wright density [2507.23467]. For \(U\sim\mathcal N(0,1)\), since \(U^2/2\sim \mathrm{Gamma}(1/2,1)\), one obtains \[ |U|\overset{\mathcal L}=|U|^\alpha X_\alpha,\qquad \alpha\in(0,1), \] where \[ \rho_{X_\alpha}(t) = 2^{(\alpha+1)/2} W_{-\alpha,(1-\alpha)/2}(-2^{\alpha-1}t^2), \qquad t>0. \] These decompositions are not merely existence statements: they characterize the exponential, gamma, and half-normal laws through functional equations for normalized Mellin transforms [2507.23467]. ## 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 \(S(\alpha)\) is a positive stable random variable of index \(\alpha\in(0,1)\), then \[ S(\alpha)^\alpha \] is a positive self-decomposable random variable. In the same framework, \[ G(\alpha+1)^{-\alpha} \] is also a positive self-decomposable random variable, where \(G(a)\) denotes a gamma random variable with parameter \(a\). These results arise from explicit transforms of Lévy exponents, \[ \psi_1(u)=u\phi(u+1), \qquad \psi_2(u)=\frac{u}{u+1}\psi_1(u+1), \] and from the factorization of the exponential law into independent positive factors [1005.4011]. Gamma laws form the main bridge between multiplicative self-decomposability and multiple selfdecomposability. For every \(t>0\), \[ \log \mathbb G_t \stackrel d= \alpha \log \mathbb G_t + T_{t,\alpha}, \] hence \[ \mathbb G_t \stackrel d= \mathbb G_t^{\,\alpha} e^{T_{t,\alpha}}. \] Thus \(\mathbb G_t\) is multiplicatively self-decomposable through the additive self-decomposability of \(\log \mathbb G_t\). More sharply, \(\log \mathbb G_t\) is twice selfdecomposable if and only if \[ t>t_1,\qquad t_1\approx 0.151649938034. \] This supplies a nontrivial threshold inside the Gamma family for higher-order multiplicative factorization on the log scale [2103.10160]. The same paper derives exact multiplicative factorizations involving weighted geometric products of independent Gamma variables. If \(\alpha=(\alpha_1,\dots,\alpha_n)\in(0,1)^n\) with \(\sum_{k=1}^n \alpha_k=1\), and \[ d(\alpha):=\prod_{k=1}^n \alpha_k^{\alpha_k}, \] then for independent Gamma\((t)\) variables \(\mathbb G_t,\mathbb G_{1,t},\dots,\mathbb G_{n,t}\): \[ \mathbb G_t \stackrel d= d(\alpha)\,\mathbb G_{1,t}^{\alpha_1}\cdots \mathbb G_{n,t}^{\alpha_n}\,e^{-X_{\alpha,t}}, \qquad t\ge \frac12, \] while for \(0