A bivariate drift and minorization bound geometric coupling error

Run the two coordinates independently until both lie in $D$. At each such visit, use the minorization to toss a common Bernoulli($ϵ$) split: on success draw one shared $\nu$ state; on failure draw independently from $R_x,R_y$. Let $T=\inf \{k+1:\text{the split at time }k\text{ succeeds}\}$, with $\inf \mathrm{\varnothing }=\mathrm{\infty }$. Record success with a measurable binary flag and evolve synchronously thereafter, without testing equality before success. The shared transition is the pushforward of $Q(X_k,\cdot )$ under $z\mapsto (z,z)$; this map is measurable on every measurable space. This construction has the required $Q$-marginals by [F1], [F2], and [F5].

Let $N_k$ count failed split transitions completed by time $k$, with $N_0=0$, and set $Z_k=a^{-k}{B}^{-N_k}h(X_k,Y_k)1_{\{T>k\}}$. If $ϵ=1$, a visit to $D^2$ couples on the next transition, so the uncoupled contribution there is zero. If $0<ϵ<1$, off $D^2$ the conditional expectation of the next $h$ is at most $ah$; on $D^2$ the uncoupled transition has probability $1-ϵ$ and expected $h$ equal to $R_x\otimes R_y(h)$. The definition of $B$ bounds its contribution after the extra factor $B^{-1}$ by $ah(x,y)$, since $h\ge 1$. Thus $(Z_k)$ is a nonnegative supermartingale in both cases.

For any integer $j\ge 1$, the probability of at least $j$ failed attempts is at most $(1-ϵ{)}^j$. On the event $T>k$ with at most $j-1$ failures, $h\ge 1$ and $B^{N_k}\le B^{j-1}$, so the supermartingale bound gives $\mathrm{Pr}(T>k)\le (1-ϵ{)}^j+a^k{B}^{j-1}h(x,y)\mathrm{.}$

Choose $\theta >0$ so $a{B}^{\theta }<1$ and set $j=\max (1,\lfloor \theta k\rfloor )$. Both terms in step 3.1 are bounded by a constant times $r^k$ for some $r<1$. Apply the coupling inequality [F3] to obtain the asserted geometric bound after increasing $M$ to cover the finite initial values of $k$.