--- title: Mittag-Leffler Markov Chain Models url: https://www.emergentmind.com/topics/mittag-leffler-markov-chain type: topic --- # Mittag-Leffler Markov Chain Models Searching arXiv for the specified Mittag-Leffler Markov-chain literature and related process papers. The Mittag-Leffler Markov chain appears in several non-equivalent Markovian constructions in which the Mittag-Leffler law enters through marginal distributions, innovation laws, jump kernels, or one-point distributions of a semigroup. In discrete time, one writes \(Z_n=\Phi Z_{n-1}+\varepsilon_n\) with \(|\Phi|<1\) and either prescribes \(ML(\alpha,\sigma)\) marginals or takes \(ML(\alpha,\sigma)\) innovations, where the two-parameter Mittag-Leffler law has Laplace transform \[ \phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0 \] [2601.06610]. In continuous time, related models include a strictly increasing chain with discrete Mittag-Leffler jumps generated by a Bernstein-Laplace operator and subordinated to a fractional Poisson clock [2010.00546], and the Mittag-Leffler process of Möhle, a time-homogeneous Feller process on \([0,\infty)\) whose marginal \(X_t\) is Mittag-Leffler distributed with parameter \(e^{-t}\) [1410.7354]. ## 1. Autoregressive discrete-time constructions A basic discrete-time Mittag-Leffler Markov chain is obtained from an AR(1) recursion with coefficient \(\Phi\), \(|\Phi|<1\). Two natural variants are distinguished. In the first, one imposes stationary marginals \(Y_n\sim ML(\alpha,1)\) and writes \[ Y_n=\Phi Y_{n-1}+\varepsilon_n,\qquad n\ge 1. \] Stationarity implies \[ \phi_Y(s)=\phi_Y(\Phi s)\phi_\varepsilon(s),\qquad \phi_Y(s)=\frac{1}{1+s^\alpha}, \] hence \[ \phi_\varepsilon(s)=\frac{1+(\Phi s)^\alpha}{1+s^\alpha}. \] Via a Bromwich contour, the innovation density is \[ f_\varepsilon(x)=\frac{1}{\pi}\int_0^\infty e^{-xy}\,y^\alpha \sin(\pi\alpha)(1-\Phi^\alpha)\, \bigl[1+2y^\alpha\cos(\pi\alpha)+y^{2\alpha}\bigr]^{-1}\,dy. \] In the second variant, the innovations themselves are Mittag-Leffler: \[ X_n=\Phi X_{n-1}+\varepsilon_n,\qquad \varepsilon_n\sim ML(\alpha,\sigma)\ \text{i.i.d.} \] The one-step conditional law then satisfies \[ P(X_n\le x\mid X_{n-1}=y)=F_\varepsilon(x-\Phi y), \] where \[ F_\varepsilon(u)=1-\frac{\sin \pi\alpha}{\pi}\int_0^\infty y^{\alpha-1}e^{-uy} \bigl[1+2y^\alpha\cos \pi\alpha+y^{2\alpha}\bigr]^{-1}\,dy,\qquad u>0. \] The first construction has ML marginals and non-ML innovations; the second has ML innovations and non-ML marginals [2601.06610]. ## 2. Transition kernels, stationarity, and moment structure For either AR(1) construction, the one-step kernel can be written as \[ P(Z_n\in dx\mid Z_{n-1}=y)=f_\varepsilon(x-\Phi y)\,dx, \] and therefore \[ P(Z_n\le x\mid Z_{n-1}=y)=\int_0^{x-\Phi y} f_\varepsilon(u)\,du. \] These expressions are closed in the sense that they involve one-dimensional integrals against elementary kernels, or inversion of known Laplace transforms [2601.06610]. Under \(|\Phi|<1\), existence of a unique strictly stationary solution of \[ Y_n=\Phi Y_{n-1}+\varepsilon_n \] follows from the usual contraction argument if \(E[\log^+|\varepsilon_1|]<\infty\). The innovation law is heavy tailed, but it has finite fractional moments of order \(\beta<\alpha\). More precisely, for \(M\sim ML(\alpha,\sigma)\), \[ E[M^p]<\infty \quad \text{iff} \quad p<\alpha. \] Since \(0<\alpha\le 1\), the law has no finite integer moments above order \(\lfloor \alpha\rfloor=0\). The AR(1) chain inherits the same no-integer-moments property, although it is geometrically ergodic under \(|\Phi|<1\) [2601.06610]. This sharp separation between ergodicity and classical moment finiteness is one of the characteristic features of the discrete-time Mittag-Leffler AR(1) chain. It also explains why moment-based inference is structurally fragile in this setting. ## 3. Empirical Laplace-transform estimation Parameter estimation for the AR(1) Mittag-Leffler chain is formulated through empirical Laplace transforms rather than sample moments. Given observations \((Z_0,\dots,Z_n)\), define \[ \phi_n(s)=\frac{1}{n}\sum_{t=1}^n e^{-sZ_t}. \] The procedure matches \(\phi_n\) to the theoretical Laplace-transform identities of the model [2601.06610]. When the observed chain has ML marginals, the identity \[ \phi_Z(s)=\phi_Z(\Phi s)\,[1+(\sigma s)^\alpha]^{-1} \] implies, at each \(s\), \[ \frac{\phi_n(s)}{\phi_n(\Phi s)}\approx [1+(\sigma s)^\alpha]^{-1}. \] Taking logs gives \[ \log \phi_n(\Phi s)-\log \phi_n(s)=\alpha\log(\sigma s). \] When the observed chain has ML marginals but non-ML innovations, one instead uses \[ \frac{\phi_n(s)}{\phi_n(\Phi s)}\approx \phi_\varepsilon(s) =\frac{1+(\Phi s)^\alpha}{1+s^\alpha}. \] Again one takes logs and fits \(\alpha,\Phi\) from two or more abscissae \(s_1,s_2\). Concretely, for \(00, \] and define the generator \[ g(\mathcal L)=\phi(\mathcal L)=\frac{\mathcal L^\alpha}{\lambda+\mathcal L^\alpha}. \] The resulting process is then time-changed by an independent fractional Poisson clock of rate \(\xi>0\) and order \(\beta\in(0,1]\), with Caputo derivative \[ {}^C D_t^\beta f(t)=\frac{1}{\Gamma(1-\beta)}\int_0^t (t-s)^{-\beta}f'(s)\,ds \] [2010.00546]. If \(p_n(t)=\Pr\{\mathcal N(t)=n\}\), the forward equation is \[ {}^C D_t^\beta p_n(t)=-\xi\,(g(\mathcal L)p)_n(t) =-\xi\sum_{m=0}^n [g(\mathcal L)]_{m,n}\,p_m(t),\qquad p_n(0)=\delta_{n0}. \] In operator form, \[ {}^C D_t^\beta \mathbf p(t)=-\xi\,\mathbf p(t)\,g(\mathcal L),\qquad \mathbf p(0)=(1,0,0,\dots). \] The solution is expressed through the Mittag-Leffler matrix function: \[ \mathbf p(t)=\mathbf p(0)\,E_\beta\bigl(-\xi t^\beta g(\mathcal L)\bigr), \qquad E_\beta(Z)=\sum_{m=0}^\infty \frac{Z^m}{\Gamma(\beta m+1)}. \] Its generating function \[ P(u,t)=\sum_{n=0}^\infty p_n(t)u^n \] satisfies \[ P(u,t)=E_\beta\!\Bigl(-\xi t^\beta \frac{(\lambda+1)(1-u)^\alpha}{\lambda+(1-u)^\alpha}\Bigr). \] The same process admits a subordination representation through the waiting-time density of the fractional Poisson clock, \[ \chi_\beta(t)=\xi\,t^{\beta-1}E_{\beta,\beta}(-\xi t^\beta), \qquad \widetilde\chi_\beta(s)=\frac{\xi}{\xi+s^\beta}, \] and the Montroll-Weiss series reproduces the Caputo equation [2010.00546]. Under the well-scaled diffusion limit \(h\to 0\), with \(\lambda=\lambda_0 h^\alpha\), \[ \mathcal L\to hD_x,\qquad h^{-\alpha}\mathcal L^\alpha\to D_x^\alpha,\qquad D_x^\alpha D_x^{-\alpha}=I, \] and \[ \lim_{h\to 0}\frac{g(\mathcal L)}{g(1)}=\mathcal G(x)=\delta(x)-\lambda_0 x^{\alpha-1}E_{\alpha,\alpha}(-\lambda_0 x^\alpha). \] The rescaled state density \(\mathcal P(x,t)\) solves \[ {}^C D_t^\beta \mathcal P(x,t) =-\xi\int_0^x \mathcal G(x-y)\mathcal P(y,t)\,dy,\qquad \mathcal P(x,0)=\delta(x). \] Equivalently, with \[ \mathcal G(x)=\delta(x)-\mathcal W_{ML,\alpha}(x),\qquad \mathcal W_{ML,\alpha}(x)=\lambda_0 x^{\alpha-1}E_{\alpha,\alpha}(-\lambda_0 x^\alpha), \] one has \[ {}^C D_t^\beta \mathcal P(x,t) =-\xi\,\mathcal P(x,t) +\xi\int_0^x \mathcal W_{ML,\alpha}(x-y)\mathcal P(y,t)\,dy. \] After integration in \(x\), these kernels coincide with the Prabhakar integrals \[ \mathfrak E_x^\gamma[f](x)=\int_0^x (x-\tau)^{\gamma-1} E_{\alpha,\gamma}[-\lambda_0(x-\tau)^\alpha]\,f(\tau)\,d\tau, \] which makes the link to the three-parameter Prabhakar fractional calculus [2010.00546]. ## 6. Möhle’s Mittag-Leffler process, scaling limits, and distinctions Möhle’s Mittag-Leffler process \(X=(X_t)_{t\ge 0}\) is a time-homogeneous Feller process on \([0,\infty)\) with the property that \(X_t\) is Mittag-Leffler distributed with parameter \(e^{-t}\), \(t\in[0,\infty)\). Its entire moments are \[ E[X_t^m]=\frac{\Gamma(1+m)}{\Gamma(1+m e^{-t})},\qquad m\in\mathbb N_0, \] and its Laplace transform is \[ E[e^{-sX_t}] =\sum_{k=0}^\infty \frac{(-s)^k}{\Gamma(1+k e^{-t})} =E_{e^{-t},1}(-s). \] The transition kernel is defined by \[ p(t,x,dy)=\mathrm{Law}\bigl(x^{e^{-t}\eta_t}\bigr), \] where \(\eta_t\) is Mittag-Leffler\((e^{-t})\), and the semigroup acts as \[ T_t f(x)=E\bigl[f(x^{e^{-t}\eta_t})\bigr]. \] Standard Feller-process theory yields a càdlàg Markov process with this semigroup and with \(X_0=1\) almost surely [1410.7354]. The infinitesimal generator admits a power-series representation. For suitable \(f\in C_0([0,\infty))\) with convergent Taylor expansions, \[ Af(x)=\lim_{t\to 0}\frac{T_t f(x)-f(x)}{t} =\sum_{k=1}^\infty \frac{f^{(k)}(x)}{k!}\,a_k(x), \] where \[ a_1(x)=x\psi(2)-x\ln x,\qquad a_k(x)=(-x)^k\ \text{for}\ k\ge 2. \] Thus \[ Af(x)=f'(x)\bigl(x\psi(2)-x\ln x\bigr) +\sum_{k=2}^\infty \frac{f^{(k)}(x)}{k!}(-x)^k. \] The finite-dimensional distributions have explicit joint moments in terms of Gamma-ratios, and the main scaling-limit result states that if \(N_t^{(n)}\) denotes the block counting process of the Bolthausen-Sznitman \(n\)-coalescent, then \[ X_t^{(n)}=\frac{N_t^{(n)}}{n e^{-t}} \] converges in \(D([0,\infty))\), the Skorohod topology of càdlàg paths, to \(X\) as \(n\to\infty\) [1410.7354]. A recurrent source of confusion is the relation between this process and the discrete-time autoregressive constructions. Möhle explicitly notes that, although one sometimes sees in the literature an “autoregressive Mittag-Leffler process” in discrete time, the process \(X\) considered here has finite moments of all orders and is of a different type. He also states that \(X\) is not a Lévy process, since its increments are neither independent nor stationary [1410.7354]. This distinguishes the coalescent-scaling-limit process sharply from the heavy-tailed AR(1) chains, whose defining feature is precisely the coexistence of geometric ergodicity with only fractional moments. Source: https://www.emergentmind.com/topics/mittag-leffler-markov-chain