Mittag-Leffler Markov Chain Models

• Mittag-Leffler Markov chains are stochastic models that incorporate the two-parameter Mittag-Leffler law, resulting in heavy-tailed distributions with fractional moment properties.
• They are constructed via autoregressive AR(1) frameworks with either ML marginals or innovations, yielding closed-form Laplace transforms and geometric ergodicity despite the absence of finite integer moments.
• Practical insights include the use of empirical Laplace-transform estimation and simulation studies, with applications ranging from high-frequency trading to fractional Poisson processes.

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

$$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$

(Dhull, 10 Jan 2026). 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 (Michelitsch et al., 2020), 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 $e^{-t}$ (Möhle, 2014).

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 $|\Phi|<1$0 and writes

$|\Phi|<1$1

Stationarity implies

$|\Phi|<1$2

hence

$|\Phi|<1$3

Via a Bromwich contour, the innovation density is

$|\Phi|<1$4

In the second variant, the innovations themselves are Mittag-Leffler: $|\Phi|<1$5 The one-step conditional law then satisfies

$|\Phi|<1$6

where

$|\Phi|<1$7

The first construction has ML marginals and non-ML innovations; the second has ML innovations and non-ML marginals (Dhull, 10 Jan 2026).

2. Transition kernels, stationarity, and moment structure

For either AR(1) construction, the one-step kernel can be written as

$|\Phi|<1$8

and therefore

$|\Phi|<1$9

These expressions are closed in the sense that they involve one-dimensional integrals against elementary kernels, or inversion of known Laplace transforms (Dhull, 10 Jan 2026).

Under $ML(\alpha,\sigma)$0, existence of a unique strictly stationary solution of

$ML(\alpha,\sigma)$1

follows from the usual contraction argument if $ML(\alpha,\sigma)$2. The innovation law is heavy tailed, but it has finite fractional moments of order $ML(\alpha,\sigma)$3. More precisely, for $ML(\alpha,\sigma)$4,

$ML(\alpha,\sigma)$5

Since $ML(\alpha,\sigma)$6, the law has no finite integer moments above order $ML(\alpha,\sigma)$7. The AR(1) chain inherits the same no-integer-moments property, although it is geometrically ergodic under $ML(\alpha,\sigma)$8 (Dhull, 10 Jan 2026).

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 $ML(\alpha,\sigma)$9, define

$ML(\alpha,\sigma)$0

The procedure matches $ML(\alpha,\sigma)$1 to the theoretical Laplace-transform identities of the model (Dhull, 10 Jan 2026).

When the observed chain has ML marginals, the identity

$ML(\alpha,\sigma)$2

implies, at each $ML(\alpha,\sigma)$3,

$ML(\alpha,\sigma)$4

Taking logs gives

$ML(\alpha,\sigma)$5

When the observed chain has ML marginals but non-ML innovations, one instead uses

$ML(\alpha,\sigma)$6

Again one takes logs and fits $ML(\alpha,\sigma)$7 from two or more abscissae $ML(\alpha,\sigma)$8. Concretely, for $ML(\alpha,\sigma)$9, one solves

$$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$0

by ordinary least-squares or direct root-finding.

Because the Mittag-Leffler law lacks ordinary integer moments in the heavy-tailed range considered here, this method is explicitly moment-free. A plausible implication is that the transform domain is not merely computationally convenient but structurally aligned with the model class.

4. Simulation evidence and high-frequency trading application

The simulation study in the AR(1) framework uses $$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$1 replications with chain length $$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$2. In the ML-innovations scenario, with $$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$3, $$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$4 or $$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$5, and $$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$6 or $$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$7, the reported averages are precise. For $$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$8, the average estimates over $$\phi_{ML}(s)=E[e^{-sM}]=\frac{1}{1+(\sigma s)^\alpha}, \qquad 0<\alpha\le 1,\ \sigma>0$$9 runs are

$[0,\infty)$0

with

$[0,\infty)$1

For $[0,\infty)$2,

$[0,\infty)$3

with

$[0,\infty)$4

In the ML-marginals scenario, with $[0,\infty)$5 and $[0,\infty)$6, similar Monte Carlo shows $[0,\infty)$7 and $[0,\infty)$8 in $[0,\infty)$9 and $X_t$0 across four combinations (Dhull, 10 Jan 2026).

A real-data illustration re-analyzes the inter-arrival times, in seconds, of positive and negative ticks in crude-oil futures HFT data with $X_t$1 records. A preliminary Poisson-process exponential fit shows large systematic departures in log-survival plots. Fitting the AR(1) version with ML-innovations yields

$X_t$2

for up-ticks, and

$X_t$3

for down-ticks. Residual diagnostic checks—Q-Q plots against $X_t$4, Kolmogorov-Smirnov $X_t$5-values $X_t$6, and Ljung-Box tests on transformed residuals—confirm an excellent fit. The AR coefficient $X_t$7–$X_t$8 captures mild short-term correlation, and the heavy-tail index $X_t$9 is consistent with fractional-Poisson waiting-time phenomena noted in the literature (Dhull, 10 Jan 2026).

5. Strictly increasing continuous-time chains with discrete Mittag-Leffler jumps

A different Mittag-Leffler Markov construction is a strictly increasing continuous-time Markov chain on $e^{-t}$0, formulated through a Bernstein-function generator. Let $e^{-t}$1 be the backward shift, $e^{-t}$2, and set the discrete Laplacian

$e^{-t}$3

Choose

$e^{-t}$4

and define the generator

$e^{-t}$5

The resulting process is then time-changed by an independent fractional Poisson clock of rate $e^{-t}$6 and order $e^{-t}$7, with Caputo derivative

$e^{-t}$8

(Michelitsch et al., 2020).

If $e^{-t}$9, the forward equation is

$\Phi$0

In operator form,

$\Phi$1

The solution is expressed through the Mittag-Leffler matrix function: $\Phi$2 Its generating function

$\Phi$3

satisfies

$\Phi$4

The same process admits a subordination representation through the waiting-time density of the fractional Poisson clock,

$\Phi$5

and the Montroll-Weiss series reproduces the Caputo equation (Michelitsch et al., 2020).

Under the well-scaled diffusion limit $\Phi$6, with $\Phi$7,

$\Phi$8

and

$\Phi$9

The rescaled state density $|\Phi|<1$0 solves

$|\Phi|<1$1

Equivalently, with

$|\Phi|<1$2

one has

$|\Phi|<1$3

After integration in $|\Phi|<1$4, these kernels coincide with the Prabhakar integrals

$|\Phi|<1$5

which makes the link to the three-parameter Prabhakar fractional calculus (Michelitsch et al., 2020).

6. Möhle’s Mittag-Leffler process, scaling limits, and distinctions

Möhle’s Mittag-Leffler process $|\Phi|<1$6 is a time-homogeneous Feller process on $|\Phi|<1$7 with the property that $|\Phi|<1$8 is Mittag-Leffler distributed with parameter $|\Phi|<1$9, $|\Phi|<1$00. Its entire moments are

$|\Phi|<1$01

and its Laplace transform is

$|\Phi|<1$02

The transition kernel is defined by

$|\Phi|<1$03

where $|\Phi|<1$04 is Mittag-Leffler$|\Phi|<1$05, and the semigroup acts as

$|\Phi|<1$06

Standard Feller-process theory yields a càdlàg Markov process with this semigroup and with $|\Phi|<1$07 almost surely (Möhle, 2014).

The infinitesimal generator admits a power-series representation. For suitable $|\Phi|<1$08 with convergent Taylor expansions,

$|\Phi|<1$09

where

$|\Phi|<1$10

Thus

$|\Phi|<1$11

The finite-dimensional distributions have explicit joint moments in terms of Gamma-ratios, and the main scaling-limit result states that if $|\Phi|<1$12 denotes the block counting process of the Bolthausen-Sznitman $|\Phi|<1$13-coalescent, then

$|\Phi|<1$14

converges in $|\Phi|<1$15, the Skorohod topology of càdlàg paths, to $|\Phi|<1$16 as $|\Phi|<1$17 (Möhle, 2014).

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 $|\Phi|<1$18 considered here has finite moments of all orders and is of a different type. He also states that $|\Phi|<1$19 is not a Lévy process, since its increments are neither independent nor stationary (Möhle, 2014). 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.