Non-uniform bounds for short asymptotic expansions in the CLT for balls in a Hilbert space
Let \(X_1,X_2,\dots\) be i.i.d. random elements with values in a separable Hilbert space \(H\), \(\dim H\in[12,\infty]\), with \(EX_1= 0\), \(E\| X_1\|^2= 1\), finite \(\beta_4= E\| X_1\|^4\) and covariance operator \(V\). Let \(Y\) denote the limiting Gaussian random element of normalized sums \(S_n= n^{-1/2}(X_1+\cdots+ X_n)\). Denote \(\Delta_n(a, r)=|P(\| S_n- a\|< r)- P(\| Y- a\|< r)- b_1|\) the error in the short expansion, where \(b_1= n^{-1/2}F_1(r^2)\) is the first term in an asymptotic Edgeworth type expansion of \(P(\| S_n- a\|< r)\). It is proved that \[ \Delta_n(a, r)\leq cn^{-1}(\beta_4+ (E(a,X_1))^4)(1+ (Va, a))(1+ |r-\| a\|\,|)^{-4}. \] The constant \(c\) depends only on 12 largest eigenvalues of \(V\). The accuracy of the estimation becomes better as \(|r- h\| a\|\,|\). No smoothness conditions are required. The estimate is best possible in some sense. It improves that of \textit{F. Götze} and \textit{V. V. Ulyanov} [Theory Probab. Appl. 51 (to appear)].
scientific article; zbMATH DE number 5078281