ON HILBERTIAN STOCHASTIC PROCESSES: CONVERGENCE IN THE CENTRAL LIMIT THEOREM

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 $\left(X_t\right)_{t \in \mathbb{Z}}$ 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 $S_n=n^{-1 / 2} \sum_{t=1}^n X_t$ in a separable Hilbert space $H$. Thus through a Fourier and multiplicative inequality of characteristic functions coupling method, we establish a convergence speed of $\left.\left.n^{(1-d) / 2}, \forall d \in\right] 1 ; 2\right]$ for the linear process. For the nonlinear process, the convergence speed is of order $n^{-1 / d} \log ^\gamma(n)$, $\gamma>0$ and $d \in] 2 ; 3]$ by a Götze-Jirak approach method. Received: June 3, 2024Revised: July 21, 2024Accepted: August 1, 2024