Functional Central Limit Theorems in Banach Spaces

Abstract

Suppose that $(X_{nj})$ is a triangular array of random variables taking values in a Banach space $E$ and that $(B_n)$ is the corresponding sequence of random paths in $E$. Conditions are considered under which the distributions of $B_n$ converge to a Gaussian measure on $C(\lbrack 0, 1 \rbrack; E)$. Under stronger conditions on the array it is shown that if $E$ is of type 2 the paths enjoy certain regularity properties, which are reflected in the convergence. The technique here is to factorise the integration procedure by which one passes from the array to the sequence of paths, using fractional integrals.