Large-deviation probabilities in Banach spaces (Q1781263)

Let \(F\), \(B\), \(F\subseteq B\), be a pair of separable Banach spaces such that \(|x|_F\geq|x|_B\) for every \(x\in F\). For independent centered pre-Gaussian random variables \(X_1,\dots, X_n\) with values in \(F\) denote \(S_n= (X_1+\cdots+ X_n)/\sqrt{n}\) and suppose covariances of \(X_k\)'s coincide with that of a centered Gaussian random variable \(Y\) with values in \(F\). Estimates for large deviation probabilities \(P(|S_n- d_n v|_B> r)\) and \(P(|S_n- d_n v|_B< r)\), where \(d_n\to \infty\), \(r_0< r< r_n\), \(v\in F\), \(|v|_B= 1\), are expressed in terms of \(P(|Y- d_n v|_B> r)\) and \(P(|Y- d_n v|_B< r)\), respectively. A technique applied in the proofs was earlier developed by \textit{V. Bentkus} and \textit{A. Rachkauskas} [Probab. Theory Relat. Fields 86, No. 2, 131--154 (1990; Zbl 0678.60005)] for the i.i.d. case and \(v=0\).