Ergodic Infinitely Divisible Stationary Process

• Ergodic infinitely divisible stationary processes are stochastic models whose finite-dimensional distributions are infinitely divisible and exhibit strict stationarity driven by ergodic flows.
• They are constructed using ID random measures integrated along measure-preserving transformations, often yielding heavy-tailed marginals and long-range dependence.
• Limit theorems, including functional central limit and sample autocovariance convergence, characterize their non-Gaussian, self-similar behavior.

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 (Owada et al., 2012, Owada, 2013, Avraham-Re'em et al., 25 Jan 2026).

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 (Owada et al., 2012, Owada, 2013).

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 (Passeggeri et al., 2017).

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 (Owada, 2013). 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 (Owada et al., 2012, Owada, 2013).

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 $x\mapsto \rho(x,\infty)$ (Owada et al., 2012).

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 $\beta\uparrow 1$ (Owada et al., 2012).

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

$c_n \in \mathrm{RV}_{\beta + 2(1-\beta)/\alpha}$

(Owada, 2013). 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 (Owada, 2013).

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 (Avraham-Re'em et al., 25 Jan 2026).

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 (Avraham-Re'em et al., 25 Jan 2026, Passeggeri et al., 2017). The ergodicity–weak mixing equivalence (Passeggeri et al., 2017) 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 (Passeggeri et al., 2017)

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 (Owada, 2013).

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 (Owada, 2013).

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 $L^2$ (Handa, 2013).

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 (Handa, 2013).

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 (Owada et al., 2012, Owada, 2013).