Approximation and exponential inequalities for sums of dependent random vectors (Q1598479)

An extension of \textit{R. C. Bradley}'s result on coupling of dependent real-valued random variables by independent ones [Mich. Math. J. 30, 69-81 (1983; Zbl 0531.60033)] is obtained for random vectors in \(\mathbb{R}^d\). It allows to establish the exponential inequalities for partial sums of \(\alpha\)-dependent random vectors in \(\mathbb{R}^d\) or their transformations to \(\mathcal{Y}\) where \(\mathcal{Y}\) is a separable Banach or Hilbert space. The Bernstein-type inequalities for partial sums of \(\beta\)-dependent random vectors with values in \(\mathcal{Y}\) are proved as well. Thus the generalizations of some results by \textit{I. F. Pinelis} and \textit{A. I. Sakhanenko} [Theory Probab. Appl. 30, 143-148 (1986); translation from Teor. Veroyatn. Primen. 30, No. 1, 127-131 (1985; Zbl 0568.60028)] and \textit{I. F. Pinelis} [ibid. 35, No. 3, 605-607 (1990), resp. ibid. 35, No. 3, 592-594 (1990; Zbl 0713.62042)] are provided for dependent summands. Several applications are given, including estimation of convergence rates in the SLLN for absolutely regular processes with values in \(\mathcal{Y}\).