## Limit theorems for empirical characteristic functionals in Banach space.(Russian)Zbl 0669.60013

The paper deals with estimates of rate of convergence of the empirical characteristic functional to the characteristic functional of a measure on a separable Banach space. Let $$\mu$$ be a probability measure on a separable Banach space F and let $$\xi_ 1,...,\xi_ n,..$$. be independent random elements in F with the distribution $$\mu$$. Denote by $$\phi_{\mu}$$ the characteristic functional of $$\mu$$. The empirical characteristic functional of $$\mu$$ is defined by the equality $\phi_{\mu,n}(f)=n^{-1}\sum^{n}_{k=1}\exp \{if(\xi_ k)\},\quad f\in F^*.$ By the accuracy of estimate of $$\phi_{\mu}$$ is meant the random variable $$\Delta_ n=\sup \{| \phi_{\mu,n}(f)- \phi_{\mu}(f)|:$$ $$\| f\| \leq 1\}$$. The author estimates $$\Delta_ n$$ in the case when $$\xi =T\eta$$, where $$\eta$$ is a random element in a separable Banach space B such that $$E\| \eta \|^ p<\infty$$, while T is an operator of type p, $$1<p<2$$. It is shown that in this case $$\Delta_ n=o(n^{-1+1/p})$$ a.s. and in mean. It is also investigated the question of validity of CLT for $$\exp \{if(\xi)\}- \phi_{\mu}(f)$$ considered as a random element in the Banach space of complex functions defined on the unit ball of $$F^*$$ and continuous in the topology $$\sigma (F^*,F)$$.
Reviewer: S.A.Chobanjan

### MSC:

 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60B10 Convergence of probability measures