A Coarsening of the Strong Mixing Condition

Let {Xt : t ∈ Z} be a collection of random variables defined on some probability space (Ω,F , P ). Mixing conditions provide one way to formalize the notion that these random variables are only weakly dependent on one another. There are many ways to define mixing; the monographs by Doukhan [8] and Bradley [5] list five classical definitions. The oldest and most general of these is the α-mixing condition of Rosenblatt [13, 4], also known as strong mixing. For any nonempty set of integers T , let FT ⊂ F denote the σ-field generated by the random variables {Xt : t ∈ T }. The α-mixing coefficients {αr : r ∈ N} associated with {Xt} are given by αr = sup S,T sup A∈FS,B∈FT |P (A ∩B)− P (A)P (B)| , (1.1)