×

Asymptotic expansions with non-uniform remainders in the central limit theorem in Hilbert space. (Russian. English summary) Zbl 0585.60011

Let \(X,X_ 1,X_ 2,..\). be a sequence of independent identically distributed random variables with values in a real separable Hilbert space H, \({\mathbb{E}}X=0\), \({\mathbb{E}}\| X\|^ 2_ H=1\), \(S_ n=n^{- 1/2}(X_ 1+...+X_ n).\) Let further w be a second order scalar polynomial on H, i.e. \(w(x)=<Cx,x>+<u,x>,\) \(x\in H\), where C:H\(\to H\) is a bounded symmetric operator and \(u\in H\). Asymptotic expansions of the probability \(\Pr \{w(S_ n)<t\}\) are obtained. The estimates of the remainder term are non-uniform.
Reviewer: Z.G.Gorgadze

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
PDFBibTeX XMLCite