Moment Inequalities and Gaussian Approximation for Mixing Sequences

Although the “mixing assumption” is nowadays considered as a rather restrictive condition, it is a powerful tool, which possesses strong properties such as preservation under functional transform, meaning that any measurable function of a mixing sequence is still mixing. Note that the mixing assumption is still often assumed in econometric literature, often to check the mixingale properties. Also, it is apparent that approximation by Markov chains, or m-dependent variables, has become a powerful tool in the analysis of dynamical systems. Moment inequalities whose upper bounds are expressed in terms of norms of conditional expectations lead to sharp moment inequalities in the case of alpha-dependent sequences or of strong mixing sequences. However, when we consider ρ‎-mixing and ϕ‎-mixing sequences, this way does not lead to the optimal moment inequalities and other techniques have to be implemented.