Uniform and nonuniform estimates in the CLT for Banach valued dependent random variables
type: Item
instance of: scholarly article
title: Uniform and nonuniform estimates in the CLT for Banach valued dependent random variables
zbMATH Open document ID: 0681.60012
DOI: 10.1016/0047-259X(88)90045-0
author: Yanyan Li
published in: Journal of Multivariate Analysis
publication date: 1988-01-01T00:00:00Z
review text: Let \(X_ i\), \(i\geq 1\), be a sequence of random variables with values in a separable Banach space B, \(S_ n=\sum^{n}_{i=1}X_ i.\) Let \(Y_ n\) be a Gaussian B-valued random element with the same covariance as \(S_ n\). Under some conditions on smoothness of the norm of B and conditions on the dependence between the summands \(X_ i\) in the spirit of Dvoretzky's and Burkholder's papers the uniform and non-uniform estimates of the quantity \[ \Delta_ n(x)=| P\{\| S_ n\| <x\}-P\{\| Y_ n\| <x\}| \] are given. Non-uniform estimates can be formulated as follows: \[ (*)\quad \Delta_ n(x)\leq C\cdot L_ n^{1/(3+2\delta)}(1+x^{4(1+\delta)^ 2/(3+2\delta)})^{-1} \] where \(L_ n=\sum^{n}_{i=1}E \| X_ i\|^{2+\delta}+E \| V^ 2_ n-1\|\), \(V^ 2_ n=\sum^{n}_{i=1}E_{i-1}\| X_ i\|^ 2\) and \(E_ i\) denotes the conditional expectation with respect to some \(\sigma\)-algebra \(F_ i.\) Reviewer's remarks: 1) In the references the paper of \textit{Wan Soo Rhee} and \textit{M. Talagrand} [ibid. 20, 303-320 (1986; Zbl 0606.60010)] is not included. This is strange, since the formulations of lemmas 1, 3-7 have one to one correspondence to lemmas 1-4, 6 and theorem 5 in the above mentioned paper. 2) The proof of the non-uniform estimate is incomplete: the estimate is proved when \(L_ n>2^{-(3+2\delta)}\) and in the case when \(L_ n<2^{-(3+2\delta)}\) and \(1+x^{\alpha}>L_ n^{- 2(1+\delta)/(3+\delta)};\) the case when \(L_ n<2^{-(3+2\delta)}\) and x is small is not considered. It is easy to see that this case could not be proved in the paper, since in the first step of the proof the author uses the estimate \[ \Delta_ n(x)\leq P\{\| S_ n\| >x\}+P\{\| Y_ n\|)>x\}. \] Here the right hand side of this inequality for fixed small x does not tend to zero as \(n\to \infty\). Thus from this inequality we can not obtain (*), which yields the estimate \[ \sup_{x}\Delta_ n(x)\leq C L_ n^{1/3+2\delta}\to 0\quad if\quad L_ n\to 0. \] 3) Reading of the paper is difficult due to many misprints; for example…
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 4115618
zbMATH Keywords: Non-uniform estimates
zbMATH Keywords: conditional expectation
reviewed by: V. Paulauskas
MaRDI profile type: Publication
cites work: On the rate of convergence to normality for sums of dependent random variables
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full work available at URL: 0047-259x(88)90045-0
OpenAlex ID: W2067519040
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description: scientific article; zbMATH DE number 4115618