A variant of an infinite-dimensional local limit theorem

Abstract

Let ξ_n be a sequence of independent, identically distributed random elements in a separable Banach space X, for which the CLTholds: the normalized sums (ξ₁+...+ξ_n)/n^1/2 converge weakly to the Gaussian random element ζ. It is proved that, under certain conditions on the distribution of ξ₁ and on the measurable mappingf: X→ R¹, the distribution of the random variable$f\left( {\frac{{\xi _1 + ... + \xi _n }}{{\sqrt n }}} \right)$ converges in variation to the distribution of the variablef(ζ).