Nonuniform Sequential Doeblin Minorization

• Nonuniform Sequential Doeblin Minorization is a framework that verifies nonuniform, blockwise minorization conditions to analyze convergence in inhomogeneous Markov processes.
• It establishes sequential geometric ergodicity by enforcing regeneration and coupling events along variable-length blocks instead of using a global minorization constant.
• The approach underpins limit theorems and mixing properties in random and nonconservative environments, aiding robust simulation and statistical estimation in complex systems.

Nonuniform Sequential Doeblin Minorization refers to a collection of minorization conditions for Markov processes, semigroups, and dynamical systems where uniform (global-in-time) minorization fails, but sufficiently strong minorization can be enforced along a nonuniform, possibly random or blockwise sequence of steps. This paradigm generalizes classical Doeblin minorization, enabling effective ergodic analysis of non-homogeneous, inhomogeneous, or random environments by constructing “blockwise” regeneration or coupling events. The concept is instrumental in establishing convergence rates, mixing properties, and limit theorems in stochastic processes and dynamical models with time-dependent, random, or sequentially accessible structure.

1. Definition and Foundational Principles

Nonuniform sequential Doeblin minorization arises when a uniform minorization condition cannot be established over all states or times. Instead, minorization is verified at (potentially irregular) times or blocks. Formally, for an inhomogeneous Markov chain $(X_j)_{j\geq0}$ with transition kernels %%%%1%%%%, a nonuniform sequential minorization condition requires, for each $j$, an integer $\ell_j > 0$, constant $\gamma_j \in (0,1)$, and probability measure $m_j$ such that

$$P_{j-\ell_j, \ell_j}(x, \Gamma) \geq \gamma_j m_j(\Gamma), \quad \forall x, \Gamma,$$

where $P_{j-\ell_j, \ell_j}$ is the transition after $\ell_j$ steps starting from time $j-\ell_j$ (Hafouta et al., 17 Oct 2025). The minorization constants $\gamma_j$ and block lengths $\ell_j$ may vary in $j$.

This condition generalizes the standard (uniform) Doeblin minorization, which requires that for all $x$ in the state space and all measurable $\Gamma$,

$P^{n_0}(x, \Gamma) \geq \gamma m(\Gamma)$

for some fixed $n_0$, $\gamma$, and $m$ (Jiang et al., 2020).

The sequential nature enables analysis in settings such as structured population dynamics, renewal-type equations, inhomogeneous Markov chains, nonconservative semigroups, and random dynamical environments, where uniform minorization is unattainable (Bansaye et al., 2017).

2. Mathematical Consequences and Geometric Ergodicity

A sequential Doeblin minorization yields nonuniform sequential geometric ergodicity. Specifically, for the concatenated transitions,

$$\sup_{x, \Gamma} |P_{j-n, n}(x, \Gamma) - A_{j,n}(\Gamma)| \leq \prod_{k=1}^{m-1}(1 - \gamma_{n_k}),$$

where $A_{j,n}$ are measures constructed via block decompositions and $n_1,\dots,n_{m-1}$ are chosen according to the sequence of minorization blocks (Hafouta et al., 17 Oct 2025).

When $\prod_{k}(1-\gamma_{n_k}) \to 0$, even at a subexponential rate (e.g., polynomial or stretched exponential), one proves sequential geometric ergodicity:

$$\sup_{x, \Gamma} |P_{j-n, n}(x, \Gamma) - \mu_j(\Gamma)| \to 0,$$

yielding effective convergence to an equilibrium measure $\mu_j$.

Such results allow for explicit contraction estimates in total variation distance for non-conservative semigroups,

$$\|\mu M_{s,t} - \mu(h_s)v(m_{s,t})\|_{TV} \leq C \mu(m_{s,t}) e^{-C_{a,B,v}(s, t)},$$

where $M_{s,t}$, $h_s$, $v$, and $C_{a,B,v}(s, t)$ are context-specific objects quantifying mass normalization and cumulative coupling capacity (Bansaye et al., 2017).

Nonuniform minorization may be realized in random environment chains, with mixing rates controlled by random variables $K(\omega)$ and decay rates $a_n$, such as

$$\|P_{\omega, n}g - \mu_{\sigma^n\omega}(g)\|_{L^\infty} \leq 2 K(\omega) a_n,$$

where $g$ is a bounded function and $\mu_{\sigma^n\omega}$ is the random equivariant (environment-dependent) stationary measure (Hafouta et al., 17 Oct 2025).

3. Role in Coupling, Block Regeneration, and Network Models

Nonuniform sequential minorization is encoded in coupling frameworks—especially Doeblin-type coupling constructions on graphs. The Doeblin graph, consisting of vertices indexed by time and state, with edges representing transition dynamics, enables analysis of merging (coupling) of paths. In the absence of uniform minorization, regeneration may occur only along certain time blocks or states (Baccelli et al., 2018).

The bridge graph $B(x^*)$—the subgraph formed by all paths initialized at a fixed state $x^*$ at all times—captures the sequential (block) regeneration structure. Unique bi-recurrent paths in $B(x^*)$ correspond to stationary processes, illustrating that sequential minorization ensures eventual merging even without uniform bounds.

These graph-theoretic models are analyzed using unimodular network techniques, enabling mass-transport arguments and simulation via finite windows and local weak convergence. For instance, the expected return time to a state and the size of time slices in the bridge graph are related via mass-transport principles.

4. Limit Theorems and Mixing in Inhomogeneous Systems

Nonuniform sequential minorization facilitates derivation of limit theorems for inhomogeneous Markov chains. Under blockwise minorization, martingale decomposition techniques yield equivalences:

• Bounded variance of sums,
• Existence of functional martingale representation,
• Berry-Esseen bounds and moderate deviation principles (Hafouta et al., 17 Oct 2025).

For functionals $S_n = \sum_{j=0}^{n-1} f_j(X_j)$ with bounded $f_j$, and variance $\sigma_n^2 = \mathrm{Var}(S_n)$, the paper establishes that central limit theorems, Wasserstein distance bounds, and moderate deviation estimates are accessible under sequential minorization (Hafouta et al., 17 Oct 2025). These results extend to chains in random dynamical environments, with mixing rates dictated by the nonuniform blockwise minorization parameters.

5. Structural and Metric Properties: Doeblin Coefficients

The structural underpinnings of nonuniform sequential minorization involve the Doeblin coefficient $\tau(W)$ of a channel or PMF collection:

$$\tau(W) = \sum_{j \in \mathcal{Y}} \min_{i \in [n]} W_{ij},$$

with complement $\gamma(P_1, \dots, P_n) = 1 - \tau(W)$ acting as a multi-way divergence (Makur et al., 2023). These coefficients enjoy tensorization,

$\tau(W \otimes V) = \tau(W)\,\tau(V),$

extremal trace characterization, and coupling interpretations, serving as metrics quantifying minimal agreement between distributions.

In sequential scenarios, the minimal component $\tau(W)$ may vary across blocks or steps, and maximal coupling probability at each block can be identified, providing operational meaning to minorization in the nonuniform setting (Makur et al., 2023). Contraction properties over Bayesian networks are formalized by recursive bounds,

$$\tau(P_{V \cup \{U\} \mid X}) \geq \tau_U\,\tau(P_{V|X}) + (1-\tau_U)\,\tau(P_{V \cup \operatorname{pa}(U) \mid X}),$$

highlighting how nonuniform minorization propagates through networked or time-dependent systems.

6. Applications in Dynamical Systems, Population Dynamics, Random Media

Sequential Doeblin minorization is crucial in contexts where mixing occurs on nonuniform scales—such as subshifts of finite type, nonhyperbolic maps, nonconservative linear PDEs, population models with time/varying parameters, and chains in random environments. In these systems, blockwise or sequential minorization allows quantification of exponential or subexponential convergence rates for occupational measures and system profiles (Kifer et al., 2012, Bansaye et al., 2017).

In statistical computation, regenerative perfect simulation algorithms for complex MCMC kernels may require artificial atoms and Bernoulli factory constructions to enforce regeneration only nonuniformly or sequentially (Lee et al., 2014).

The framework also extends to ergodic-theoretic questions of uniqueness and mixing. Blockwise minorization methods (via control of variation in g-functions) enforce uniqueness of invariant measures in chains with infinite memory, and the absence of uniform mixing may result in non-mixing despite uniqueness (Berger et al., 2023).

7. Practical and Theoretical Implications

A practical implication is that via nonuniform sequential minorization—whether verified through coupling/block decomposition, control of Doeblin coefficients, or spectral methods—one can deduce rates of mixing and limit theorems in settings where uniform minorization is unattainable. This includes inhomogeneous and random environments, nonconservative semigroups, and structured and agent-based probabilistic models.

A plausible implication is that in high-dimensional or highly variable systems, nonuniform sequential Doeblin minorization affords robust tools for establishing convergence, estimating mixing times, and constructing efficient simulation and unbiased estimation algorithms. Moreover, the theory unifies convergence analysis in Markov chains, renewal processes, linear PDEs, ergodic and dynamical systems, and complex Bayesian and graphical models, through the lens of sequential regeneration and coupling phenomena.

In conclusion, nonuniform sequential Doeblin minorization generalizes classic ergodic theory, enabling precise quantitative analysis and strong statistical theorems in heterogeneous stochastic and dynamical models where uniform mixing mechanisms fail.