} {2^{14} d\beta^4\log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}}}$, assume additionally that $n=T/\delta$ is an integer.\
Let $\bx_t$ and $\bw_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{\bx_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{\bw_0}_2^2}\leq R^2 + \beta^2/m$. Then there exists a coupling between $\bx_t$ and $\bw_t$ such that $$\begin{aligned}
            \E{f(\bx_{i\delta} - \bw_{i\delta})} \leq e^{-\lambda i\delta}  \E{f(\bx_{0} - \bw_{0})} + \frac{6}{\lambda}  \lrp{L + \LN^2} \epsilon
            \end{aligned}$$

From Lemma \[l:energy\_x\] and \[l:energy\_w\], our initial conditions imply that for all $t$, $\E{\|\bx_t\|_2^2} \leq 6\lrp{R^2 + \frac{\beta^2}{m}}$ and $\E{\|\bw_{k\delta}\|_2^2} \leq 8 \lrp{R^2 + \frac{\beta^2}{m}}$.

Consider an arbitrary $k$, and for $t\in[0,T)$, define $$\begin{aligned}
            x_t := \bx_{kT+t} \quad \text{and} \quad
            w_t := \bw_{kT+t} \numberthis \label{e:t:kasjnd}
            \end{aligned}$$ Notice that as described above, $x_t$ and $w_t$ have dynamics described in and . Let $x_t,w_t$ have joint distribution as described in and , and let $(y_t,v_t)$ be the processes defined in and . Notice that the joint distribution between $x_t$ and $w_t$ equivalently describes a coupling between $\bx_t$ and $\bw_t$ over $t\in[kT, (k+1)T)$.

First, notice that the processes $\eqref{e:coupled_4_processes_x}$ and $\eqref{e:coupled_4_processes_y}$ have the same distribution as . We can thus apply Lemma \[l:gaussian\_contraction\]: $$\begin{aligned}
            \E{f(x_{T} - y_{T})}
            \leq& e^{-\lambda T} \E{f(x_0 - y_0)} + 6T (L+\LN^2) \epsilon
            \end{aligned}$$

By Lemma \[l:non\_gaussian\_contraction\_stationary\], $$\begin{aligned}
            \E{f(x_T - v_T)} - \E{f(x_T - y_T)}
            \leq 4TL\epsilon
            \end{aligned}$$ By Lemma \[l:non\_gaussian\_contraction\_anisotropic\], $$\begin{aligned}
            \E{f(x_T - w_T)} - \E{f(x_T - v_T)}
            \leq 4T(L+\LN^2)\epsi