Non-iid Martingale Difference Sequences

• Non-iid martingale difference sequences are stochastic processes with zero conditional expectation but varying distributions and dependencies.
• They underpin advanced statistical inference by enabling applications of stable CLTs in time series, dynamic causal studies, and Monte Carlo simulations.
• Techniques like Bernstein blocking and lag-adjusted structures simplify proofs of stable convergence in the presence of heterogeneity and serial dependence.

A non-iid sequence of martingale difference type refers to a stochastic process (or array) whose increments (differences) are martingale differences—meaning each term has zero conditional expectation given the relevant filtration—but where the increments are not required to be identically distributed nor independent. In modern probability theory, and especially in applications involving time series, random fields, and Monte Carlo estimation, such non-iid martingale difference sequences represent a highly flexible class for both modeling and inference, accommodating a broad range of dependence structures and heterogeneity.

1. Definition and Structure

A sequence $\{d_k\}_{k\geq 0}$ is a martingale difference sequence with respect to a filtration $\{\mathcal{F}_k\}$ if

$$\mathbb{E}[d_k\,|\,\mathcal{F}_{k-1}] = 0 \qquad \text{for each } k.$$

The “non-iid” qualifier indicates that the distributions of the $d_k$ may vary with $k$, both in law and conditional variance, and no independence (over $k$) is assumed, only the martingale difference property.

This generality is crucial for modeling: in time-series or high-dimensional settings, the increments may naturally display time-varying or spatially heterogeneous behavior while still exhibiting martingale difference characteristics—for example, in GARCH or ARCH processes, processes with non-stationary volatility, or iterated random function Markov chains (Dedecker et al., 2014, Benhenni et al., 2020).

In contrast to i.i.d. sequences, martingale difference sequences allow for extensive serial dependence as long as (conditional) orthogonality is maintained.

2. Central Limit Theorems: Multivariate and Lagged Extensions

Stable CLTs for non-iid martingale difference arrays and sequences generalize the classical martingale central limit theorem by establishing weak (or stable) convergence of suitably normalized sums to a Gaussian (or sometimes mixed Gaussian) limit under “conditional” moment and covariance requirements.

Let $\{X_{n,k}\}$ be a triangular array of $\mathbb{R}^q$-valued random vectors adapted to filtrations $\{\mathcal{F}_{n,k}\}$ with the property

$\mathbb{E}[X_{n,k} \mid \mathcal{F}_{n,k-1}] = 0.$

The array is a lag-$p$ martingale difference array (Huch et al., 7 Oct 2025) if the differences are only conditionally mean-zero with respect to a delayed (“lagged”) filtration, i.e., $\mathbb{E}[X_{n,k}|\mathcal{F}_{n,k-p}] = 0$.

Key conditions for stable CLT (Häusler et al., 26 Jul 2024, Huch et al., 7 Oct 2025):

• Conditional Lindeberg condition: For all $\epsilon>0$,

$$\sum_k \mathbb{E}\big[\|X_{n,k}\|^2 \mathbf{1}_{\{\|X_{n,k}\|>\epsilon\}}\,|\,\mathcal{F}_{n,k-1}\big] \xrightarrow{P} 0$$

• Conditional covariance convergence: There exists a random, symmetric, positive semidefinite matrix $\Psi$ (possibly $\mathcal{G}$-measurable for some $\sigma$-field $\mathcal{G}$) such that

$\sum_k X_{n,k} X_{n,k}^T \xrightarrow{P} \Psi$

These conditions allow for heterogeneity (non-constant variance) and temporal dependence structures.

Multivariate extension and Cramér–Wold Device:

To show the $q$-variate array $\{X_{n,k}\}$ converges stably (in law) to $\Psi^{1/2} Z_q$, where $Z_q\sim N(0,I_q)$ independent of $\mathcal{G}$, it suffices to verify for all nonzero $u\in\mathbb{R}^q$: $$\left\langle u, S_n \right\rangle = \sum_k \langle u, X_{n,k} \rangle \xrightarrow{stably} (u^T \Psi u)^{1/2} Z.$$ Once this univariate CLT holds for all $u$, multivariate stable convergence follows by the Cramér–Wold device (Huch et al., 7 Oct 2025).

3. Methodological Advances: Bernstein Blocking and Lag-Adjusted Structures

When classical martingale difference structure is too restrictive—such as in settings where conditional expectation zero only holds for lagged filtrations (e.g., in dynamic causal inference)—modern techniques employ Bernstein blocking and lag-adjusted filtrations:

• Bernstein Blocking: Partition the data into blocks so that within-block dependencies are negligible in the limit; analyze the central limit behavior blockwise, then aggregate to recover the overall limiting behavior.
• Lag Martingale Differences: Processes where

$\mathbb{E}[X_k\,|\,\mathcal{F}_{k-p}]=0,$

but possibly $\mathbb{E}[X_k\,|\,\mathcal{F}_{k-1}]\neq 0$. Such arrays generalize the applicability of martingale CLTs to processes with more complex temporal dependencies (Huch et al., 7 Oct 2025).

This broadens the martingale CLT toolkit for high-dimensional or dependent processes arising in, for example, dynamic treatment regime analysis or time-varying coefficient models.

4. Generalization via Cramér–Wold and Stable Convergence

The key theoretical result for vector-valued non-iid martingale differences is that multivariate stable convergence can be deduced from all one-dimensional projections:

• For each $u \in \mathbb{R}^q \setminus \{0\}$, define $Y_{n,k} = \langle u, X_{n,k} \rangle$.
• Verify that the Lindeberg-type and variance conditions hold for each projected sequence $\{Y_{n,k}\}$.
• Prove univariate (scalar) stable CLT for each projection.
• Conclude multivariate stable convergence $$S_n \equiv \sum_k X_{n,k} \xrightarrow{stably} \Psi^{1/2}Z_q$$ by the Cramér–Wold theorem (Huch et al., 7 Oct 2025, Häusler et al., 26 Jul 2024).

Thus, the analytic focus reduces to (potentially simpler) univariate verification, dramatically simplifying proofs in the non-iid martingale context.

5. Practical Applications

The general theory of non-iid martingale difference sequences and their stable central limit behavior underpins a wide variety of statistical and machine learning methodologies:

• Dynamic causal inference: Stable CLTs for lagged martingale difference arrays enable rigorous inference in time-varying, dynamically influenced systems, where independence is not present but “sequential ignorability” with respect to a lagged filtration holds (Huch et al., 7 Oct 2025).
• Nonparametric regression and time series: Estimating means or other functionals under dependent, heteroscedastic, or time-varying error structures capitalizes on stable CLT results for such arrays (Benhenni et al., 2020).
• Monte Carlo simulation and stochastic optimization: Stopping rules and error estimation for sample means in noisy (non-iid) martingale difference regimes crucially depend on implementable CLT-based error quantification (Wu et al., 26 Oct 2025).

6. Simulation Studies and Robustness

Simulation studies show that the theoretical stable CLT results accurately describe finite-sample performance in settings with lagged dependence and non-homogeneous variance. The practical reliability of batch mean estimates based on non-iid martingale differences has been validated by empirical studies demonstrating proper coverage and error rate control in sequential Monte Carlo estimation under such general conditions (Wu et al., 26 Oct 2025, Huch et al., 7 Oct 2025).

7. Summary Table: Key Results and Methods

Paper	Main Concept	Key Technical Tool
(Häusler et al., 26 Jul 2024)	Stable CLT for multivariate martingales	Conditional Lindeberg + covariance, Cramér–Wold
(Huch et al., 7 Oct 2025)	Lag-martingale, stable CLT	Bernstein blocking + Cramér–Wold
(Wu et al., 26 Oct 2025)	Stopping rules, Monte Carlo	Martingale CLT, practical variance estimation

Conclusion

Non-iid sequences of martingale difference type form a mathematically robust and highly applicable class of stochastic processes. Stable central limit theorems for such sequences (including lagged variants) under conditional Lindeberg-type and covariance conditions provide the theoretical foundation for statistical inference in complex, temporally dependent, and heterogeneous data environments. The Cramér–Wold device, Bernstein blocking, and careful variance normalization are fundamental methods enabling such limit results, with direct impacts from theoretical probability to practical data analysis in domains ranging from high-dimensional learning to sequential Monte Carlo and causal inference (Huch et al., 7 Oct 2025, Häusler et al., 26 Jul 2024, Wu et al., 26 Oct 2025).