On Berry Esseen type estimates for randomized Martingales in the non stationary setting
J Dedecker, F Merlevède, M Peligrad, Vishakha Sharma
TL;DR
This work extends Berry-Esseen-type Gaussian approximation results to randomized weighted sums of non-stationary martingale differences, via the Principle of Conditioning. By projecting onto a random vector $\theta$ uniform on the unit sphere and comparing $S_n(\theta)=\sum_j \theta_j d_j$ to a Gaussian $N_{\theta}$ with conditional variance $\sum_j \theta_j^2 \mathbb{E}(d_j^2)$, the authors derive an $O((1+v_n)/n)$ bound on the Kolmogorov distance in expectation, where $v_n$ encodes higher-moment and variance-deviation terms. Under decorrelation-type second-order conditions, they also obtain an alternative $O(((\alpha_n,\beta_n,\sigma_4^2(n))\log n)/n)$-typebound and, in a nonstationary ARCH example, demonstrate a rate of $(\log n)^2/n$, illustrating faster convergence than classical non-stationary Berry-Esseen bounds. The methodology—combining the Principle of Conditioning with independent-case Berry-Esseen bounds and sphere concentration—offers a robust framework for Gaussian approximation of randomized martingales in non-stationary settings with potential applications to statistics with random designs or projections.
Abstract
In this paper, we consider partial sums of triangular martingale differences weighted by random variables drawn uniformly on the sphere, and globally independent of the martingale differences. Starting from the so-called principle of conditioning and using some arguments developed by Klartag-Sodin and Bobkov-Chistyakov-G{ö}tze, we give some upper bounds for the Kolmogorov distance between the distribution of these weighted sums and a Normal distribution. Under some conditions on the conditional variances of the martingale differences, the obtained rates are always faster than those obtained in case of usual partial sums.