---
title: Ergodic Infinitely Divisible Stationary Process
url: https://www.emergentmind.com/topics/ergodic-infinitely-divisible-stationary-process
type: topic
---

# Ergodic Infinitely Divisible Stationary Process

An ergodic infinitely divisible stationary process is a class of stochastic processes where each finite-dimensional marginal is infinitely divisible, the process is strictly stationary, and the underlying measure-preserving dynamical system is ergodic. Such processes are typically constructed via integral representations over ergodic, conservative flows, and may feature heavy-tailed marginals or long-range dependence, with the interplay of infinite divisibility and ergodic-theoretic properties determining both probabilistic and limit behaviors [1209.3957][1302.0058][2601.17988].

## 1. Construction and Canonical Representation

Let $(E,\mathcal E, \mu)$ be a $\sigma$-finite (often infinite) measure space. An infinitely divisible (ID) process $\{X_n\}_{n\ge1}$ is defined via an ID random measure $M$ over $(E,\mathcal E)$ with control measure $\mu$ and symmetric (or non-symmetric) Lévy measure $\rho$ on $\mathbb R$. For a measurable function $f: E\rightarrow\mathbb R$, the process is given by
\[
X_n = \int_E f(T^n x)\,M(dx),
\]
where $T: E \rightarrow E$ is a measure-preserving (often conservative and ergodic) transformation [1209.3957][1302.0058].

The marginal distributions are symmetric, infinitely divisible, and typically heavy-tailed, with $\rho(x,\infty)\in \mathrm{RV}_{-\alpha}$ for $0<\alpha<2$. The spectral (Lévy–Khintchine) representation elucidates the full law of any finite block $(X_{n_1},\ldots,X_{n_k})$ through a characteristic function parameterized by drift, Gaussian, and jump components, as in the full Lévy-Khintchine formula [1704.02503].

## 2. Ergodicity and Conservative Infinite-Measure Flows

The ergodicity of an ID stationary process is inherently tied to the dynamical properties of the flow $T$. For infinite measure $\mu$, conservativity—where orbits of sets of finite $\mu$-mass return infinitely often—replaces classical ergodicity. The flow $T$ is pointwise dual ergodic if there exists a sequence $a_n\to\infty$, regularly varying with index $\beta\in[0,1)$, such that
\[
\frac1{a_n}\sum_{k=1}^n \widehat{T}^k h \longrightarrow \mu(h)\quad\text{almost everywhere}
\]
for any $h\in L^1(\mu)$, where $\widehat T$ is the dual operator [1302.0058]. The decomposition of $E$ into conservative and dissipative parts (Hopf’s decomposition) identifies the regime of long-range dependence (the conservative part).

Processes of the form above are ergodic (infinite-measure sense) and typically mixing if $T$ is pointwise dual ergodic and $f$ is properly supported. The memory parameter $\beta$ governs the strength of long-range dependence, with $\beta=0$ ("short" memory, dissipative), and higher $\beta$ yielding heavier dependence [1209.3957][1302.0058].

## 3. Functional Central Limit Theorem and Limit Processes

For symmetric stationary ID processes generated through conservative pointwise dual ergodic flows, the partial sum process satisfies a functional central limit theorem (FCLT) in the space $D[0,\infty)$:
\[
\frac{1}{c_n} \sum_{k=1}^{\lceil n\cdot \rceil} X_k \Rightarrow \mu(f) Y_{\alpha,\beta}(\cdot)
\]
with normalization
\[
c_n = \Gamma(1+\beta) C_\alpha^{-1/\alpha} a_n \rho^{\leftarrow}(1/w_n),
\]
where $w_n = \mu(\bigcup_{k=0}^{n-1} T^{-k}A)$ and $\rho^{\leftarrow}$ is the left-inverse function of $x\mapsto \rho(x,\infty)$ [1209.3957].

The limit $Y_{\alpha,\beta}$ is a class of symmetric $\alpha$-stable, $H$-self-similar processes with stationary increments:
\[
Y_{\alpha,\beta}(t) = \int_{\Omega'\times[0,\infty)} M_\beta((t-x)_+, \omega')\, Z_{\alpha,\beta}(d\omega',dx)
\]
with $H = \beta + (1-\beta)/\alpha$. The finite-dimensional log-characteristic function is given by
\[
-\log \mathbb{E} \exp\Big\{i \sum_{j=1}^k \theta_j Y_{\alpha,\beta}(t_j)\Big\}
= \int_0^\infty \mathbb{E}'\bigg| \sum_{j=1}^k \theta_j M_\beta((t_j-x)_+, \omega') \bigg|^\alpha (1-\beta) x^{-\beta} dx.
\]
This class interpolates between $1/\alpha$-self-similar (short memory) and $1$-self-similar (very long memory) as $\beta\uparrow 1$ [1209.3957].

## 4. Limit Theory for Sample Autocovariances and Long Memory

The sample autocovariance
\[
\hat\gamma_n(h) = \frac1n \sum_{t=1}^{n-h} X_t X_{t+h}
\]
exhibits asymptotics determined by both tail index $\alpha$ and memory parameter $\beta$. Under appropriate normalization,
\[
\bigg(\frac{\hat\gamma_n(h)}{c_n},\, h=0,\ldots, H\bigg) \Rightarrow \left(\mu(f\,f\circ T^h)\,W,\, h=0,\ldots, H\right)
\]
where $W$ is a positive strictly stable $\alpha/2$ random variable, and
\[
c_n \in \mathrm{RV}_{\beta + 2(1-\beta)/\alpha}
\]
[1302.0058]. For $\beta=0$ (dissipative/short memory), $c_n\sim n^{2/\alpha}$, recovering classical heavy-tailed moving-average scaling. For $\beta\uparrow 1$, the exponent approaches $1$, indicating increased long-range dependence [1302.0058].

Sample autocorrelations converge in probability (the random $W$ cancels),
\[
\hat\rho_n(h) \xrightarrow{p} \frac{\mu(f\,f\circ T^h)}{\mu(f^2)},\quad h\ge0
\]
demonstrating robust asymptotic predictability even in infinite-variance regimes.

## 5. Ergodicity, Weak Mixing, and Mixing Properties

For any group $G$, every ergodic, infinitely divisible, stationary process that is separable in probability is automatically weakly mixing. This holds regardless of any topological structure on $G$. The core result is:
If $X = (X_g)_{g\in G}$ is a stationary ID $G$-process, ergodic and separable in probability, then the product system $X\otimes X$ is also ergodic, i.e., $X$ is weakly mixing [2601.17988].

Proof techniques reduce to Poissonian representations (Maruyama-type), extensions to Polish groups, and the ergodicity criterion for Poisson suspensions. The argument applies to symmetric $\alpha$-stable processes $(0<\alpha<2)$ and more generally to all stationary separable-in-probability ID processes on countable or Polish group indices [2601.17988][1704.02503]. The ergodicity–weak mixing equivalence [1704.02503] is also established using Fourier-analytic spectral representations.

Sufficient conditions for mixing and ergodicity include:
- Decay of the Gaussian covariance kernel
- Vanishing of the Lévy measure’s small-jump coupling at large lags
- Spectral factorization of bivariate characteristic functions at high separation [1704.02503]

## 6. Notable Examples and Model Classes

**Null-recurrent Markov shifts:** Let $T$ be the left shift on a null-recurrent Markov chain’s path space; then $T$ is conservative and pointwise dual ergodic, leading to an ergodic ID stationary process driven by heavy-tailed Lévy measure, generalizing earlier work on stable integrals [1302.0058].

**Piecewise-smooth interval maps with indifferent fixed points:** Interval maps admitting neutral fixed points and infinite absolutely continuous invariant measures drive examples where the memory parameter $\beta$ exactly quantifies long-range dependence [1302.0058].

**α-CIR Models:** Ergodic, stationary, ID processes on $\mathbb R_+$ or measure-valued spaces (measure-valued branching with immigration) can be generated using the α-CIR framework, with jump mechanisms governed by stable laws. For strictly positive drift $b(r)$, these models possess a unique stationary infinitely divisible law and a positive spectral gap in $L^2$ [1307.2407].

**Generalized Fleming–Viot processes:** Time-changed ratios of independent α-CIRs yield measure-valued processes with unique stationary laws inherited from the ergodic α-CIR structure. Such processes demonstrate algebraic decay of variance even in the absence of a spectral gap [1307.2407].

## 7. The Interplay of Tail and Memory Parameters

The asymptotic behavior of ergodic ID stationary processes is regulated jointly by the tail index $\alpha$ of the Lévy measure and the memory parameter $\beta$ from the ergodic flow. The scaling exponents for limit theorems on partial sums and autocovariances combine both parameters:
\[
\text{Scaling index for autocovariances: }\;\; \beta + 2(1-\beta)/\alpha,
\]
\[
\text{Limiting process self-similarity: } H = \beta + (1-\beta)/\alpha.
\]
This suggests that longer memory (large $\beta$) dramatically alters the stochastic scaling, promoting heavy long-range dependence, and that all limit processes---in the heavy-tailed, infinite-variance case---are non-Gaussian, stable, and self-similar in a highly non-classical fashion [1209.3957][1302.0058].

Source: https://www.emergentmind.com/topics/ergodic-infinitely-divisible-stationary-process