Sharp equivalence between ρ- and τ-mixing coefficients

Abstract

For A and B two σ-algebras, the ρ-mixing coefficient ρ(A, B) between A and B is the supremum correlation between two real random variables X and Y being resp. A-and B-measurable; the τ (A, B) coefficient is defined similarly, but restricting to the case where X and Y are indicator functions. It has been known for long that the bound ρ Cτ (1 + |log τ |) holds for some constant C; in this article, I show that C = 1 fits and that that value cannot be improved. |Cov(X, Y)| Var(X) 1/2 Var(Y) 1/2 (where the supremum is taken only for non-constant X and Y). This coefficient is 0 if and only if A and B are independent; and we will say that A and B are as correlated (in the ρ-mixing sense) as ρ(A, B) is large. Note that one always has ρ(A, B) 1, because of the Cauchy-Schwarz inequality. There are other ways to measure dependence between A and B (see for instance the review paper [2]): in particular, rather than looking at correlation between A-and B-measurable random variables, we can look at correlation between events. The most classical measure of dependence in this category is the α-mixing coefficient: (2) α(A, B) := sup

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A comprehensive extension of the FGM copula

“…In particular, the copula by Cuadras and Augé (1981) and the Chogosov copula, studied by Peyre (2013), are extended herewith to comprehensive families. In Sect.…”

A "cousin" of a theorem of Csáki and Fischer

“…Here we shall state his result in the notations used here in this paper. (The notations used by Peyre [20] slightly conflict with those used here. )…”

“…Corollary 5 (trivial embellishment of Peyre's [20] example). For any t ∈ (0, 1) and any ε > 0, there exist a probability space (Ω, F , P ) and σ-fields A and B ⊂ F with the following properties: (i) there exist events A ∈ A and…”

“…An example is constructed to show a limitation of this "cousin". Also, this "cousin" is used to trivially embellish a very sharp example of Peyre [20] in connection with the comparison of these two measures of dependence. Suppose (Ω, F , P ) is a probability space.…”