Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable central limit theorems for discrete-time lag martingale difference arrays

Walter Dempsey, Easton Huch

TL;DR

This work introduces lag martingale difference sequences and extends central limit theorems to both scalar and vector forms with sublinear and diverging lags. By employing Bernstein blocking, the authors achieve 𝒢-stable convergence to Gaussian limits with random variances, enabling valid inference in dynamic causal settings where carryover effects exist. The theory is specialized to dynamic causal inference frameworks like Bojinov–Rambachan–Shephard, and a simulation study demonstrates the necessity of incorporating the random limiting variance via BarΨ for accurate inference. Overall, the results provide a practical, weak-assumption pathway to asymptotically Gaussian inference in complex time-dependent treatments and outcomes.

Abstract

Recent work in dynamic causal inference introduced a class of discrete-time stochastic processes that generalize martingale difference sequences and arrays as follows: the random variates in each sequence have expectation zero given certain lagged filtrations but not given the natural filtration. We formalize this class of stochastic processes and prove a stable central limit theorem (CLT) via a Bernstein blocking scheme and an application of the classical martingale CLT. We generalize our limit theorem to vector-valued processes via the Cramér-Wold device and develop a simple form for the limiting variance. We demonstrate the application of these results to a problem in dynamic causal inference and present a simulation study supporting their validity.

Stable central limit theorems for discrete-time lag martingale difference arrays

TL;DR

This work introduces lag martingale difference sequences and extends central limit theorems to both scalar and vector forms with sublinear and diverging lags. By employing Bernstein blocking, the authors achieve 𝒢-stable convergence to Gaussian limits with random variances, enabling valid inference in dynamic causal settings where carryover effects exist. The theory is specialized to dynamic causal inference frameworks like Bojinov–Rambachan–Shephard, and a simulation study demonstrates the necessity of incorporating the random limiting variance via BarΨ for accurate inference. Overall, the results provide a practical, weak-assumption pathway to asymptotically Gaussian inference in complex time-dependent treatments and outcomes.

Abstract

Recent work in dynamic causal inference introduced a class of discrete-time stochastic processes that generalize martingale difference sequences and arrays as follows: the random variates in each sequence have expectation zero given certain lagged filtrations but not given the natural filtration. We formalize this class of stochastic processes and prove a stable central limit theorem (CLT) via a Bernstein blocking scheme and an application of the classical martingale CLT. We generalize our limit theorem to vector-valued processes via the Cramér-Wold device and develop a simple form for the limiting variance. We demonstrate the application of these results to a problem in dynamic causal inference and present a simulation study supporting their validity.

Paper Structure

This paper contains 26 sections, 11 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lag martingale difference sequences and arrays
  3. Basic CLT
  4. Setup
  5. Theorem statement
  6. Basic CLT proof
  7. Convergence in distribution.
  8. Proof of Theorem \ref{['thm:stable-clt']}.
  9. Multivariate extension
  10. Multivariate definitions
  11. Multivariate CLT
  12. Multivariate CLT proof
  13. Diverging lags
  14. Diverging lag definition
  15. Blocking sequences
  16. ...and 11 more sections

Key Result

Proposition 1

Let $p \in \{0, 1, \ldots\}$, $q \in \{p+1, p+2, \ldots\}$, and $n \in \mathbb{N}$ such that $k_n \geq q+1$. If $(X_{nk})_{p+1 \leq k \leq k_n}$ is a lag-$p$ martingale, then $(X_{nk})_{q+1 \leq k \leq k_n}$ is a lag-$q$ martingale.

Figures (2)

  • Figure 1: Estimates from the simulation study. Panel (a) plots kernel density estimates (KDEs) of the sampled values of $\tau$ and the corresponding unbiased estimator $\Hat{\tau}$. Panel (b) compares a KDE of the sampled values of $W \coloneqq \Hat{\tau} - \tau$ to a Gaussian distribution with $\sigma_W \coloneqq \mathrm{SD}(W)$, offering visual evidence of the non-Gaussianity of $W$. Panel (c) shows the estimates of $\Bar{\psi}$ segmented by whether the 100th value of $A_t=1$ is even or odd, highlighting how the limiting variance is a random variable.
  • Figure 2: Normalized estimates of $W$ from the simulation study. Panel (a) plots a KDE of $Z \coloneqq \left(T / \Bar{\psi}\right)^{1/2} W$ vs. a standard Gaussian distribution, showing close agreement as expected based on \ref{['eq:mixing']}. Panel (b) performs a similar comparison but omitting the covariance terms in the definition of $\Bar{\psi}$; we label the resulting quantity $Z$-naive. The comparison of the KDE for $Z$-naive to the standard Gaussian distribution shows significant disagreement, highlighting the need for the covariance terms to obtain valid asymptotic inference.

Theorems & Definitions (24)

  • Definition 1: Lag-$p$ martingale difference sequence
  • Proposition 1: Lag relationships
  • proof
  • Theorem 1: CLT
  • Lemma 1: CLT
  • proof
  • Definition 2: Multivariate lag-$p$ martingale difference sequence
  • Theorem 2: Multivariate CLT
  • Definition 3: Lag martingale difference sequence
  • Proposition 2: Diverging blocks
  • ...and 14 more