Asymptotically precise estimate of the accuracy of Gaussian approximation in Hilbert space. (English) Zbl 0675.60011
Let \(X_ 1,X_ 2,..\). be i.i.d. random variables on a separable Hilbert space H, with E \(X_ 1=0\), E \(| X_ 1|^ 3<\infty\). Denote by V the covariance operator of \(\sigma^{-1}X_ 1\), where \(\sigma^ 2=E | X_ 1|^ 2\) and let Y stand for a (0,V) Gaussian H-valued random variable. The paper contains an asymptotically precise estimate of
\[
P(| (\sigma \sqrt{n})^{-1}\sum^{n}_{1}X_ i-a| <r)- P(| Y-a| <r),
\]
for all \(a\in H\), \(r\geq 0\), depending on the first six eigenvalues of V. The proof uses F. Goetze’s approach [Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 333-355 (1979; Zbl 0405.60009)] to the estimation of the characteristic function of \(| \sum^{n}_{1}X_ i|^ 2\) and some ideas due to V. V. Yurinskij [Teor. Veroyatn. Primen. 27, 270-278 (1982; Zbl 0565.60005); English translation in Theory Probab. Appl. 27, 280-289 (1982)].
Reviewer: V.Sakalauskas
MSC:
| 60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |
| 60E10 | Characteristic functions; other transforms |
References:
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