Coupling for τ-Dependent Sequences and Applications

Let X be a real-valued random variable and $$M$$ a σ-algebra. We show that the minimum $${\mathbb{L}}^1$$ -distance between X and a random variable distributed as X and independant of $$M$$ can be viewed as a dependence coefficient τ( $$M$$ ,X) whose definition is comparable (but different) to that of the usual β-mixing coefficient between $$M$$ and σ(X). We compare this new coefficient to other well known measures of dependence, and we show that it can be easily computed in various situations, such as causal Bernoulli shifts or stable Markov chains defined via iterative random maps. Next, we use coupling techniques to obtain Bennett and Rosenthal-type inequalities for partial sums of τ-dependent sequences. The former is used to prove a strong invariance principle for partial sums.