Uniform and nonuniform estimates in the CLT for Banach valued dependent random variables
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 in (4.1) the exponents \(-2+2\delta\) and \(2+\delta\) must be -2-2\(\delta\) and \(2+2\delta\), respectively; in the right hand side of (4.6) the factor \((1+x^{\alpha})^{-2(1+\delta)/\alpha}\) is missing.
scientific article; zbMATH DE number 4115618
scientific article