Total variation bounds in the Lindeberg central limit theorem
N. T. Dung, H. T. P. Thao
TL;DR
This paper addresses the problem of obtaining an explicit total-variation bound in the central limit theorem for sums of independent, non-i.i.d. random variables. It develops a Stein's-method-based approach, leveraging Stein kernels and score functions to bound $d_{TV}(S_n,N)$ in terms of $E[|X_k|^2\min(b_n,|X_k|)]/b_n^3$ and the Fisher-information-based quantities $J(X_k)$. A key contribution is showing that, under a uniform bound on $J(X_k)$, Lindeberg's condition is equivalent to TV convergence of $S_n$ to $N$ together with the Feller-Lévy condition, thus strengthening the classical CLT to convergence in total variation under minimal moment assumptions. The results also extend to cases where Gaussian noise is added to the summands, preserving TV convergence and highlighting the role of standardized Fisher information in controlling the rate of convergence.
Abstract
In this paper, we obtain an explicit total variation bound in the central limit theorem for the sums of non-i.i.d. random variables. Our results show that, under suitable assumptions, Lindeberg's condition is sufficient and necessary for the convergence in total variation distance.