In this paper the analogue in infinite dimensions of the Erdos-Dvoretzky rate of escape test for finite dimensional Brownian motion is proved. Some examples are constructed which exhibit the essential differences between the finite and infinite dimensional cases and which suggest several conjectures and problems.
"Rates of Escape of Infinite Dimensional Brownian Motion." Ann. Probab. 8 (2) 325 - 338, April, 1980. https://doi.org/10.1214/aop/1176994780