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Article Dans Une Revue Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Année : 2022

Rates of convergence in the central limit theorem for martingales in the non stationary setting

Résumé

In this paper, we give rates of convergence, for minimal distances and for the uniform distance, between the law of partial sums of martingale differences and the limiting Gaussian distribution. More precisely, denoting by $P_{X}$ the law of a random variable $X$ and by $G_{a}$ the normal distribution ${\mathcal N} (0,a)$, we are interested by giving quantitative estimates for the convergence of $P_{S_n/\sqrt{V_n}}$ to $G_1$, where $S_n$ is the partial sum associated with either martingale differences sequences or more general dependent sequences, and $V_n= {\rm Var}(S_n)$. Applications to linear statistics, non stationary $\rho$-mixing sequences and sequential dynamical systems are given.
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Dates et versions

hal-03112369 , version 1 (16-01-2021)

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Jérôme Dedecker, Florence Merlevède, Emmanuel Rio. Rates of convergence in the central limit theorem for martingales in the non stationary setting. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2022, 58 (2), ⟨10.1214/21-aihp1182⟩. ⟨hal-03112369⟩
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