Abstract
If $f = \{f_t\mid t \in T\}$ is a centered, second-order stochastic process with bounded sample paths, it is then known that $f$ satisfies the central limit theorem in the topology of uniform convergence if and only if the intrinsic metric $\rho^2_f$ (on $T$) induced by $f$ is totally bounded and the normalized sums are eventually uniformly $\rho^2_f$-equicontinuous. We show that a centered, second-order stochastic process satisfies the central limit theorem in the topology of uniform convergence if and only if it has bounded sample paths and there exists totally bounded pseudometric $\rho$ on $T$ so that the normalized sums are eventually uniformly $\rho$-equicontinuous.
Citation
N. T. Andersen. V. Dobric. "The Central Limit Theorem for Stochastic Processes." Ann. Probab. 15 (1) 164 - 177, January, 1987. https://doi.org/10.1214/aop/1176992262
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