Bentkus, V. Yu.; Zalesskij, B. A. Asymptotic expansions with non-uniform remainders in the central limit theorem in Hilbert space. (Russian. English summary) Zbl 0585.60011 Litov. Mat. Sb. 25, No. 3, 3-16 (1985). 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 Cited in 3 Reviews MSC: 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) Keywords:central limit theorem; bounded symmetric operator; Asymptotic expansions; estimates of the remainder term PDFBibTeX XMLCite \textit{V. Yu. Bentkus} and \textit{B. A. Zalesskij}, Litov. Mat. Sb. 25, No. 3, 3--16 (1985; Zbl 0585.60011)