In this paper, we deal with a problem in probability theory, namely, the law convergence of Hilbertian processes. We first consider a causal stochastic process  in Wold representation. In the second step, we assume a nonlinear process under a strong mixing condition. We develop new methods to evaluate the convergence of the normalized partial n-sum  in a separable Hilbert space H. Thus through a Fourier and multiplicative inequality of characteristic functions coupling method, we establish a convergence speed of  for the linear process. For the nonlinear process, the convergence speed is of order   and  by a Götze-Jirak approach method.

characteristic functions, covariance operator, Hilbert spaces, Fourier-Stieltjes transform, rate of convergence, Central Limit Theorem

Received: June 3, 2024; Revised: July 21, 2024; Accepted: August 1, 2024; Published: September 12, 2024

How to cite this article: Claude YAMEOGO, Victorien F. KONANE and Wahabo BAGUIAN, On Hilbertian stochastic processes: convergence in the Central Limit Theorem, Far East Journal of Theoretical Statistics 68(3) (2024), 305-333. https://doi.org/10.17654/0972086324018

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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