Strong-mixing covariance bound in terms of tail quantiles

For a finite real $U$, write $T(t)=\mathbb{P}(\mathrm{\mid }U\mathrm{\mid }>t)$ for $t\ge 0$. The tail tends to zero at infinity, is nonincreasing and is right-continuous. Thus $Q_U(u)$ is finite for $u>0$, is nonincreasing, and is Borel measurable. More precisely, $Q_U(u)>t$ if and only if $u<T(t)$: if $T(t)\le u$, then $t$ belongs to the set defining the infimum; if $T(t)>u$, right-continuity supplies an $\epsilon >0$ with $T(t+\epsilon )>u$, so that infimum exceeds $t$. Consequently Lebesgue measure on $(0,1)$ gives $\lambda \{u:Q_U(u)>t\}=T(t)$ for every $t\ge 0$. The laws of $Q_U$ under $\lambda$ and of $\mathrm{\mid }U\mathrm{\mid }$ therefore agree on all closed left rays and both have total mass one. These rays form a pi-system and generate the Borel sigma-algebra: their differences give half-open intervals, and countable unions of those give all open intervals. By [F6] the laws agree, proving the stated integral identity. In particular ${\int }_0^1{Q}_U^2=\mathbb{E}{U}^2$.

Suppose first $U,V\ge 0$ and $U,V\in L^2$. Their first moments and $\mathbb{E}(UV)$ are finite by [F3]. The threshold-indicator identities and [F5] give $\mathbb{E}(UV)={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathbb{P}(U>s,V>t)dsdt$ and $\mathbb{E}U\mathbb{E}V={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\mathbb{P}(U>s)\mathbb{P}(V>t)dsdt$. Both integrands have finite integrals, so their difference represents the covariance and the integral triangle inequality applies. The absolute threshold-indicator covariance is bounded by $a$, by $\mathbb{P}(U>s)$, and by $\mathbb{P}(V>t)$, since the joint probability and the product probability both lie between zero and either marginal probability.

Put $T_U(s)=\mathbb{P}(U>s)$ and $T_V(t)=\mathbb{P}(V>t)$. By [F5] and step 1.1, ${\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\min \{a,T_U(s),T_V(t)\}dsdt={\int }_0^a{Q}_U(u)Q_V(u)du\mathrm{.}$ Indeed the minimum is ${\int }_0^a{\mathbf{1}}_{\{u<T_U(s)\}}{\mathbf{1}}_{\{u<T_V(t)\}}du$, and each inner threshold set has length its quantile. The right-hand side is finite by Cauchy–Schwarz on $(0,1)$ and the squared-quantile identity in step 1.1; this particular integral Cauchy–Schwarz inequality follows by integrating $(Q_U-c{Q}_V{)}^2\ge 0$ and minimizing the resulting quadratic, with the zero-norm case immediate. Step 1.2 therefore proves the covariance bound with constant one for nonnegative $U,V$.

For real $U,V$, expand $\mathrm{Cov}(U^{+}-U^{-},V^{+}-V^{-})$ into four terms. Each nonnegative part has a tail quantile at most the corresponding absolute quantile, because its tail probabilities are smaller. The result of step 2.1 bounds each term by ${\int }_0^a{Q}_U{Q}_V$; the triangle inequality gives the asserted factor four. If $\mathrm{\mid }U\mathrm{\mid }\le A$ and $\mathrm{\mid }V\mathrm{\mid }\le B$, their absolute quantiles are bounded by $A,B$, so the bound becomes $4ABa$, including $a=0$ or either zero bound.

For complex $U,V$, expand $\mathbb{E}(UV)-\mathbb{E}U\mathbb{E}V$ into the four real bilinear covariance terms of their real and imaginary parts. Each real part is bounded in absolute value by $A$ or $B$, so step 3.1 and the complex triangle inequality give $16ABa$. Finally, for observations separated by $n$, both collection sigma-algebras are sub-sigma-algebras of the past and future fields appearing in [F1]; their event supremum is therefore at most ${\alpha }_Y(n)$. All assertions follow.