Geometrically Ergodic Markov Chains Overview

• Geometrically ergodic Markov chains are defined by exponential convergence to equilibrium using drift and small set conditions.
• They underpin robust MCMC methods by offering explicit spectral gap properties, concentration bounds, and quantitative rate controls.
• These chains facilitate practical applications in high-dimensional sampling and statistical estimation, ensuring reliable simulation outcomes.

Geometrically ergodic Markov chains are a broad and technically foundational class within Markov process theory, characterized by exponential convergence to equilibrium when measured in an appropriately chosen norm. This concept underlies much of modern theory and practice in Markov chain Monte Carlo (MCMC), statistical estimation in dependent data, optimal transport, and the spectral theory of stochastic processes. The structure, deep equivalences with drift and spectral gap conditions, quantitative concentration, advanced perturbation bounds, and implications for algorithmic robustness position geometric ergodicity as a key concept across probability, statistics, and computational mathematics.

1. Precise Definitions and Core Characterizations

A discrete-time Markov chain $\{X_n\}$ with transition kernel $P$, state space $E$, and stationary distribution $\pi$ is geometrically ergodic if there exist a measurable function $V:E \to [1,\infty)$, constants $\lambda \in (0,1)$, $b>0$, and a “small set” $C$ such that

$PV(x) \leq \lambda V(x) + b \cdot \mathbf{1}_C(x)$

and, for some norm (typically total variation or a $V$-weighted norm), there exist $C_x < \infty$ and $0 < \rho < 1$ such that

$\|P^n(x, \cdot) - \pi\| \leq C_x \rho^n.$

Uniform versions (e.g., $V$-uniform ergodicity) impose the bound with $C_x$ replaced by $C V(x)$. These conditions are equivalent under minorization and drift assumptions (Gallegos-Herrada et al., 2022).

For irreducible aperiodic chains on general or countable state spaces, geometric ergodicity is also characterized by the existence of an exponential moment for the first return time to a small set: there exist $C, \kappa > 1$ such that

$$\sup_{x \in C} E_x\left[\kappa^{\tau_C}\right] < \infty.$$

Alternatively, spectral gap conditions or operator-norm spectral radius criteria provide formulations for geometric ergodicity, with further variants in reversible settings.

2. Drift (Lyapunov) and Small Set Conditions

Drift and minorization are the backbone of geometric ergodicity theory. The drift (Foster–Lyapunov) condition

$PV(x) \leq \lambda V(x) + b \cdot \mathbf{1}_C(x)$

ensures the chain is attracted towards a region of state space where regeneration or strong coupling is possible. The small set (minorization) condition states that for some $m \geq 1$, $\delta > 0$, and probability measure $\nu$, the $m$-step kernel obeys

$P^m(x, \cdot) \geq \delta \nu(\cdot)$

for $x \in C$. In the presence of such conditions, convergence rates and concentration inequalities can be linked directly to the explicit constants in the drift and minorization inequalities.

Higher regularity, such as differentiability or higher-order sensitivity of the invariant measure with respect to perturbations in the kernel, can also be deduced under strengthened drift or Doeblin–Fortet inequalities (Ferré et al., 2012).

3. Equivalence to Spectral Gap and Operator-Theoretic Formulations

Geometric ergodicity connects directly to spectral properties of the Markov kernel. For reversible chains, the restriction of $P$ to $L_2^0(\pi)$ (mean zero functions) having spectral radius strictly less than one is equivalent to $L_2$ geometric ergodicity. Explicitly, if $\|Pf\|_{L_2(\pi)} \leq \rho \|f\|_{L_2(\pi)}$ for some $\rho < 1$ and all mean-zero $f$, then

$$\|P^n(x, \cdot) - \pi\|_{L_2(\pi)} \leq C_{\mu} \rho^n$$

for any initial distribution $\mu$ (Negrea et al., 2017, Gallegos-Herrada et al., 2022).

Equivalences extend to operator-norm inequalities in $L_{\infty,V}$ or $L_1(\pi)$ (for total variation), continuity of the spectrum of $P$, and geometric contraction in Wasserstein distance (when paired with a suitable drift).

A major contribution is the systematic catalog of 34 equivalent conditions for geometric ergodicity, including convergence bounds, drift inequalities, spectral gap, small sets, moment conditions, and operator norms; these equivalences hold for general and reversible chains (Gallegos-Herrada et al., 2022).

4. Quantitative and Functional Consequences

Geometric ergodicity underlies powerful quantitative results. For additive functionals $S_n = \sum_{k=0}^{n-1} (g(X_k) - \pi(g))$, moment inequalities (Rosenthal-type) and Bernstein-type concentration inequalities are available: $$P_\pi(|S_n| \geq t) \leq 2 \exp\left\{ -\frac{t^2}{2 \operatorname{Var}_\pi(S_n) + J^{1/(\gamma + 3)} t^{2-1/(\gamma+3)}} \right\},$$ where $J$ is explicit in terms of mixing and drift constants (Durmus et al., 2021). The subgaussian tails for separately bounded functionals are a necessary and sufficient condition for geometric ergodicity (Dedecker et al., 2014, Havet et al., 2019).

Suitable empirical Bernstein inequalities, via tailored martingale decompositions and coupling arguments, deliver nonasymptotic error bounds for unbounded functionals and self-normalized processes (Wintenberger, 2015).

For regular variation contexts and extremes, geometric drift ensures standard central limit theorems and convergence of the tail empirical process, while its failure can yield degenerate or stable non-Gaussian limits (Kulik et al., 2015).

5. Stability, Perturbation Theory, and Algorithmic Robustness

A principal consideration in MCMC and stochastic modeling is the robustness of geometric ergodicity under kernel perturbations. Several advanced results quantify how close the invariant distribution and finite-time distributions are under small perturbations, using either total variation, $V$-weighted norms, or Wasserstein distance (Rudolf et al., 2015, Negrea et al., 2017, Ferré et al., 2012, Mao et al., 2020).

For reversible geometrically ergodic chains, if the approximating kernel $\widetilde{P}$ satisfies $$\|P - \widetilde{P}\|_{L_2(\pi)} \leq \epsilon < \alpha$$, where $\alpha$ is the spectral gap of $P$, then $\widetilde{P}$ is geometrically ergodic and

$$\|\pi - \widetilde{\pi}\|_{L_2(\pi)} \leq \frac{\epsilon}{\sqrt{\alpha^2 - \epsilon^2}}$$

(Negrea et al., 2017). The cumulative error between $n$-step distributions can be explicitly bounded in terms of the single-step perturbation, the Lyapunov function, and the geometric contraction rate (Rudolf et al., 2015).

For $V$-geometrically ergodic chains, the Keller–Liverani theorem allows stability under two-norm continuity conditions, yielding continuity and higher differentiability of the invariant measure as a function of model parameters (Ferré et al., 2012).

6. Applications to MCMC, Random Environments, and Infinite-Dimensional Models

Geometric ergodicity is essential for validating the use of MCMC estimators, establishing CLTs, and providing finite-sample confidence bounds. For instance, trans-dimensional reversible jump MCMC algorithms inherit geometric ergodicity from the within-model kernels if these are themselves geometrically ergodic; the chain's decomposition leads to explicit L² rate control (Qin, 2023).

In Markov chains with randomly varying environments, geometric drift and minorization can be adapted to depend on the environment’s state, still yielding geometric rates and ergodic theorems under weak technical conditions (Gerencser et al., 2018, Truquet, 2021). For high- or infinite-dimensional state spaces—such as function-valued processes in spatial extremes—the geometric ergodicity framework extends to Polish (but not necessarily locally compact) spaces by leveraging generalized drift and minorization (Koch et al., 2017).

7. Methodologies and Algorithmic Constructions

Verifying geometric ergodicity relies on establishing Lyapunov drift and minorization, either directly in the original chain or via reduction (e.g., examining marginal or block components as in two-variable Gibbs samplers (Tan et al., 2012)). Renewal theory tools—such as the first entrance–last exit decomposition and quantitative versions of the Kendall theorem—facilitate explicit convergence rate calculations and error analysis in MCMC output (Bednorz, 2013).

Sophisticated martingale techniques, incorporating returns to small sets, yield sharp concentration and deviation inequalities suited for MCMC output analysis (Havet et al., 2019, Wintenberger, 2015). Perturbation analyses typically use operator-theoretic expansions, spectral mapping, and coupling methods (including Nummelin splitting for non-atomic chains) to control changes in stationary distribution and convergence rates after small modifications in the transition kernel (Mao et al., 2020).

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Geometric ergodicity serves as a central linking thread between detailed operator/spectral analysis, concrete quantitative rate control, stochastic stability, and practical reliability in the simulation and inference tasks carried out with Markov chains. Its rigorous characterizations, diverse equivalent formulations, and rich perturbation theory provide a robust foundation for modern stochastic computation, statistical estimation, and the mathematical theory of dependence.