On Bernstein Type and Maximal Inequalities for Dependent Banach-Valued Random Vectors and Applications

This article discusses some results on Bernstein type and maximal inequalities for partial sums of dependent random vectors taking their values in separable Hilbert or Banach spaces of finite or infinite dimension. Two types of measure of dependence are considered: strong mixing coefficients (α-mixing) and absolutely regular mixing coefficients (β-mixing). These inequalities, which are similar to those in the dependent real case, are used to derive the strong law of large numbers (SLLN) and the bounded law of the iterated logarithm (LIL) for absolutely regular Hilbert- or Banach-valued processes under minimal mixing conditions. The article first introduces the relevant notation and definitions before presenting the maximal inequalities in the strong mixing case, followed by the absolutely regular mixing case. It concludes with some applications to the SLLN, the bounded LIL for Hilbertian or Banachian absolutely regular processes, the recursive estimation of probability density, and the covariance operator estimations.