Pruitt's Estimates in Banach Space

TL;DR

This paper extends Pruitt's estimates for exit times of random walks to infinite-dimensional Banach spaces, revealing their dependence on space type and cotype, and characterizing these properties.

Contribution

It introduces a novel extension of Pruitt's estimates to infinite-dimensional Banach spaces and characterizes space types based on these estimates.

Findings

Estimates depend on whether the space is type 2 or cotype 2.

These estimates characterize the type and cotype of Banach spaces.

Extension of finite-dimensional results to infinite-dimensional settings.

Abstract

Pruitt's estimates on the expectation and the distribution of the time taken by a random walk to exit a ball of radius r are extended to the infinite dimensional setting. It is shown that they separate into two pairs of estimates depending on whether the space is type 2 or cotype 2. It is further shown that these estimates characterize type 2 and cotype 2 spaces.