## 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

$\displaystyle (Z>u,T>u)=(U<2a,Uu),Uu) )$

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.