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

Kolchinskij, V. I.

Teor. Veroyatn. Mat. Stat., Kiev 39, 71-78 (1988).

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

Keywords:

estimates of rate of convergence of the empirical characteristic functional; measure on a separable Banach space