Nummelin Splitting Technique

• Nummelin splitting technique is a probabilistic method that augments a Markov chain's state space into a split space to explicitly generate regeneration points.
• It decomposes chain trajectories into nearly independent blocks, facilitating uniform ergodicity estimates and enabling rigorous central limit theorem and bootstrap analysis.
• The method underpins applications in ergodic BSDEs, control problems, and Gaussian approximations, ensuring robust inference for dependent Markov data.

The Nummelin splitting technique is a probabilistic construction that facilitates regeneration in Markov chains by augmenting the state space and introducing randomized transitions, resulting in an embedded structure where renewal properties become explicit. This methodology enables the decomposition of chains into independent or lightly dependent blocks, making it a cornerstone in the analysis of ergodicity, empirical process theory, and inferential procedures for dependent data. Its applications encompass proving uniform ergodicity, establishing existence and uniqueness results in ergodic backward stochastic difference equations (BSDEs), and facilitating bootstrap methods for Harris recurrent Markov chains.

1. Formal Construction and Principles of Nummelin Splitting

The foundation of Nummelin splitting lies in augmenting the original state space $𝓧$ of a Markov chain into a "split space" $𝓧̂ = 𝓧 × \{0, 1\}$, effectively creating two layers: $𝓧_0$ and $𝓧_1$ (Allan et al., 2015). Transitions in this space are governed by a splitting parameter $γ ∈ (0, 1)$ and ensure that, with probability $γ$, the chain "regenerates" from a prescribed distribution, independent of its prior trajectory.

A probability measure $v$ on $𝓧$ is split as

$𝓘(v) = \begin{pmatrix}(1-\gamma)v \ \gamma v\end{pmatrix},$

while the transition matrix splitting involves, for a perturbed matrix $B$ that is $γ$-controlled by a reference matrix $A$,

$C = \frac{B - γA}{1 - γ}.$

The augmented transition matrix then uses $C$ for transitions from $𝓧_0$ and $A$ from $𝓧_1$, with inter-layer splitting. This architecture supports the explicit construction of "renewal" or "regeneration" points.

In the context of Harris recurrent Markov chains under a minorization condition, the split chain $(X_t, Y_t)$ is governed by a Bernoulli indicator $Y_t$, which marks regeneration times when $Y_t = 1$ (Choi et al., 20 Oct 2025).

2. Regeneration Times and Block Decomposition

A principal innovation inherent in Nummelin splitting is the partition of chain trajectories into regenerative blocks. Defining a regeneration time $\sigma_0 = \min\{t ≥ 0: Y_t = 1\}$, the chain's history is decomposed into blocks:

• $B_1 = (X_1, ..., X_{\sigma_0 \cdot m})$
• $$B_2 = (X_{(\sigma_0+1)m}, ..., X_{(\sigma_0+\sigma_1)m})$$, ...

For block length $m=1$, blocks are independent; for $m > 1$, they are at worst one-dependent. A generic additive functional sums over blocks:

$$\sum_{t=0}^{n-1} [f(X_t) - \pi(f)] \approx \sum_{i=1}^I \check{f}(B_i) + R_n,$$

with $\check{f}(B_i) = \sum_{t \in B_i} f(X_t)$ and $R_n$ a remainder comprising incomplete block contributions (Choi et al., 20 Oct 2025).

This decomposition underlies advanced central limit theorems, allowing chains previously intractable due to dependence to be analyzed using independent block techniques.

3. Uniform Ergodicity Estimates

Splitting directly facilitates the establishment of uniform ergodicity results, critical for stochastic differential equations, control theory, and empirical processes. By examining two independent split chains and their first meeting time $\hat{S}$, exponential moment bounds are obtained:

$$\mathbb{E}^{x,y}\left[e^{\beta \hat{S}}\right] \leq 1 + \epsilon\quad \forall \beta \in [0, \beta_0],$$

where constants depend only on the splitting parameters and the reference kernel (Allan et al., 2015). This yields uniform estimates for total variation distances:

$\|P^t_p - \pi\|_{TV} \leq (1+\epsilon) e^{-pt},$

where $P^t_p$ is the perturbed chain at time $t$, $\pi$ is its stationary distribution, and $R$ can approach $1$ for sufficiently small $p$. These bounds hold uniformly for all kernels $B$ controlled by $A$.

Such estimates are vital for controlling solution behavior over long horizons, forming the quantitative backbone of existence and uniqueness proofs, as well as facilitating limiting arguments in ergodic control applications.

4. Applications in Ergodic BSDEs and Control Problems

In ergodic BSDEs, the uniform ergodicity provided by splitting underpins the passage from discounted to ergodic equations:

$$Y_T = Y_t - \sum_{u=t}^{T-1}\left(f(X_u, Z_u) - \lambda \right) + \sum_{u=t}^{T-1} Z_u^* M_{u+1},$$

with $\lambda$ the ergodic cost. Managing solution differences uniformly over time intervals relies on the chain "forgetting" its initial state quickly, ensured by Nummelin-based rates (Allan et al., 2015). This produces unique stationary, Markovian solutions to the EBSDE.

Within ergodic control, the objective function for the control problem is

$$J(x, U) = \limsup_{T\to\infty} \frac{1}{T} \mathbb{E}_x^U\left[\sum_{s=0}^{T-1} L(X_s, U_s)\right].$$

Uniform ergodicity (guaranteed by splitting) ensures the existence of long-run averages independent of the starting condition, facilitating the construction of Hamiltonians and verification of optimal feedback controls via EBSDE solutions.

5. Regenerative Bootstrap and Empirical Process Methods

Leveraging the regenerative blocks arising from Nummelin splitting, advanced bootstrap schemes for dependent sequences are constructed. The regenerative block bootstrap forms the basis for Gaussian multiplier bootstrapping:

$$G_n^\zeta(f) = \frac{1}{\sqrt{n}} \sum_{i=1}^{\hat{i}_n} \zeta_i \left\{ \check{f}(\widehat{B}_i) - \ell(\widehat{B}_i)\, \widehat{\pi} f \right\},$$

where $\zeta_i$ are independent standard Gaussians and $\ell(\widehat{B}_i)$ gives block length (Choi et al., 20 Oct 2025). Blocks constructed from estimated transition kernels yield a practical (approximate) split chain, extending inference even when the original dynamics are unknown.

This bootstrap is employed to construct uniform confidence bands for invariant densities (kernel density estimators):

$$I_\alpha(x) = \left[\widehat{\pi}_w(x) - \frac{\hat{c}_n(\alpha)\,\widehat{\sigma}_n(x)}{\sqrt{n}}, \, \widehat{\pi}_w(x) + \frac{\hat{c}_n(\alpha)\,\widehat{\sigma}_n(x)}{\sqrt{n}} \right],$$

where $\hat{c}_n(\alpha)$ is a bootstrap estimate of the $(1-\alpha)$ quantile of the supremum of the bootstrapped empirical process.

A plausible implication is that this approach bypasses more restrictive empirical process conditions (such as Smirnov–Bickel–Rosenblatt) and yields non-asymptotic error rates compatible with the independence case. The transformation from a dependent Markov chain to a nearly independent block structure is central in modern bootstrap methodology for ergodic Markov models.

6. Covariance Structure and Gaussian Approximation

Following Nummelin splitting, the covariance structure for Gaussian approximations of additive functionals is given by

$$\Sigma(f, g) = \beta^{-1} \left\{ \operatorname{Cov}(f(B_1), g(B_1)) + \operatorname{Cov}(f(B_1), g(B_2)) + \operatorname{Cov}(f(B_2), g(B_1)) \right\}$$

with

$$\beta = m\, \mathbb{E}_{\check{\alpha}}[\tau_{\check{\alpha}}],$$

where $m$ is block length and $\tau_{\check{\alpha}}$ the interval between regenerations (Choi et al., 20 Oct 2025). This structure allows precise control of Gaussian (and bootstrap) approximation errors, with non-asymptotic rates paralleling those for sums of high-dimensional, weakly dependent vectors.

Such control is crucial for inference involving the supremum of empirical processes indexed by function classes, including cases where classical weak convergence does not hold (non-Donsker settings).

7. Context, Methodological Role, and Implications

Nummelin splitting is recognized as a powerful methodology for studying ergodicity under perturbations, controlling the convergence of Markov chains in total variation, and facilitating coupling and renewal approaches. In the context of stochastic equations and control, it guarantees the regularity and robustness of limiting procedures, ensuring solution uniqueness and existence for ergodic BSDEs under general conditions (Allan et al., 2015). In empirical process theory and bootstrap inference, the splitting makes the strong independence assumptions tractable for dependent Markov chains, enabling modern inference for invariant measures and functional estimates (Choi et al., 20 Oct 2025).

This suggests its methodological significance spans both foundational probability theory and applied statistical inference for Markovian data. The technique supports the development of non-asymptotic and finite-sample guarantee results, which are widely applicable, so long as the requisite minorization and regeneration conditions are satisfied.

The Nummelin splitting technique has thus become an established tool for the analysis of Markov chains, ergodic processes, and inferential procedures built upon renewal structures.