A total-variation rate bounds strong-mixing coefficients

Fix $n\ge 1$, $j\ge 0$, $A\in {\mathcal{F}}_j$, and a future event $B\in \sigma (X_{j+n},X_{j+n+1},\dots )$. By [F2], for the path indicator defining $B$ there is a measurable $h:\mathsf{X}\to [0,1]$ with $\mathbb{E}[{\mathbf{1}}_B\mid {\mathcal{F}}_{j+n}]=h(X_{j+n})$. Applying [F3] and the tower property gives $\mathbb{E}[{\mathbf{1}}_B\mid {\mathcal{F}}_j]=P^{n}h(X_j)$. Stationarity gives $\mathbb{P}(B)=\pi (h)$.

For probability measures $\mu ,\nu$ and measurable $0\le h\le 1$, the dyadic simple functions $h_m=2^{-m}{\sum }_{j=1}^{2^m}{\mathbf{1}}_{\{h\ge j{2}^{-m}\}}$ differ from $h$ by at most $2^{-m}$. Their integrals differ by at most $\mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}$, since each summand is an indicator and [F4] bounds its measure difference. Letting $m\to \mathrm{\infty }$ proves $\mathrm{\mid }\mu (h)-\nu (h)\mathrm{\mid }\le \mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}$.

Apply step 1.2 with $\mu =P^n(x,\cdot )$ and $\nu =\pi$. The given rate then yields $\mathrm{\mid }{P}^{n}h(x)-\pi (h)\mathrm{\mid }\le M(x)r^n$ for every $x$.

Using [F5], [F2], and [F3], ${\displaystyle \mathrm{\mid }\mathbb{P}(A\cap B)-\mathbb{P}(A)\mathbb{P}(B)\mathrm{\mid }=\mathrm{\mid }\mathbb{E}[{\mathbf{1}}_A(P^{n}h(X_j)-\pi (h))]\mathrm{\mid }\le \mathbb{E}\mathrm{\mid }{P}^{n}h(X_j)-\pi (h)\mathrm{\mid }\le \pi (M)r^n\mathrm{.}}$ The first inequality uses $0\le {\mathbf{1}}_A\le 1$; the last follows from the preceding pointwise bound and stationarity of $X_j$. Since $A,B,j$ were arbitrary, taking the supremum in [F1] proves the claim.