On Hoeffding’s Inequality for Dependent Random Variables

Let {Z θ : θ ∈ Θ} be a random process indexed by a parameter θ in some (metric) space Θ. Given an appropriate moment or probability inequality for each fixed θ (a pointwise inequality), one can often derive an inequality that holds uniformly in θ ∈ Θ by applying the chaining technique. Therefore, pointwise inequalities are (apart from being of intrinsic interest) quite relevant within the theory of stochastic processes. We present a generalization of Ho-effding's inequality, and the related bounded difference inequality of McDiarmid [7]. We also state the corresponding uniform inequality. As an application, we consider estimation in the autoregression model.