Rio’s Inequality

Let {X} and {Y} be two integrable real-valued random variables and let { Q_x(u) = inf\{t: P(|X|>t) \leq u \}} be the quantile function of {|X|}. Then if {Q_X Q_Y} is integrable over { (0,1)} we have

\displaystyle \boxed{|Cov(X,Y)| \leq 2 \int\limits_{0}^{2a} Q_x(u) Q_Y(u) du}

where { a= a(\sigma(X), \sigma(Y)) = \sup\limits_{\substack{B \in \mathcal{B} \\ C \in \mathcal{C}}} |Cov(\mathbb{I}_{\sigma(X)},\mathbb{I}_{\sigma(Y)})|} is the a-mixing coefficient.

Proof: Set {X^{+} = sup(0,X)} and {X^{-} = sup(0,-X)} then

\displaystyle Cov(X,Y) = Cov(X^{+},Y^{+}) + Cov(X^{-},Y^{-}) - Cov(X^{+},Y^{-}) - Cov(X^{-},Y^{+})

since { X = (X^{+} - X^{-})} and { Y = (Y^{+} - Y^{-})}

note also that

\displaystyle Cov(X^+,Y^+) = \int \int_{\mathbb{R}^{2}_{+}} [P(X>u, Y> \upsilon) - P(X>u)P(Y> \upsilon)]du d\upsilon

which implies that

\displaystyle |Cov(X^+,Y^+)| \leq \int \int_{\mathbb{R}^{2}_{+}} \inf (a, P(X>u),P(Y> \upsilon))du d\upsilon

Now since

\displaystyle (\alpha \wedge a \wedge c)+ (\alpha \wedge a \wedge d) + (\alpha \wedge b \wedge c) + (\alpha \wedge b \wedge d) \leq 2[(2\alpha) \wedge (a+b) \wedge (c+d)]

\displaystyle \begin{array}{rcl} |Cov(X,Y)|& =& | Cov(X^{+},Y^{+}) + Cov(X^{-},Y^{-}) - Cov(X^{+},Y^{-}) - Cov(X^{-},Y^{+})| \\ &\leq& 2 \int \int_{\mathbb{R}^{2}_{+}} \inf (2a, P(|X|>u),P(|Y|> \upsilon))du d\upsilon =: \mathcal{I} \end{array}

Now let {U} be a r.v. with uniform distribution over {[0,1 ]} and let {(Z,T)} be a bivariate r.v. defined as

\displaystyle (Z,T) = (0,0)_{\mathbb{I}_{U \geq 2a}} + (Q_X(u),Q_Y(u) )_{\mathbb{I}_{U < 2a}}

Hence

\displaystyle E(ZT) = \int\limits_{0}^{2a}Q_X(u)Q_Y(u)du

and

\displaystyle (Z>u,T>u)=(U<2a,U<P(|X|>u),U<P(|Y|>u) )

thus

\displaystyle \begin{array}{rcl} E(ZT) &=& \int \int_{\mathbb{R}^{2}_{+}} (P(Z>u, T> \upsilon)) du d\upsilon \\ &=& \int \int_{\mathbb{R}^{2}_{+}} \inf (2a, P(|X|>u), P(|Y|> \upsilon)) du d\upsilon \end{array}

\Box

References:

D.Bosq (1996). Nonparametric Statistics for Stochastic Processes. Springer

E.Rio (1993). Covariance inequalities for strongly mixing processes. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques 29.4 : 587-597.

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