= Synchronous contraction of a random affine recursion
{title2=$\mathbb E|X_n-Y_n|=\rho^n\mathbb E|X_0-Y_0|$}
Drive two copies of a <random affine recursion> with the same coefficient pairs, independent of their initial <coupling>. Then $X_n-Y_n=A_n(X_{n-1}-Y_{n-1})$. If $\rho=\mathbb E|A_1|<1$ and $\mathbb E|X_0-Y_0|<\infty$, <independence of random variables> gives
$$
\mathbb E|X_n-Y_n|=\rho^n\mathbb E|X_0-Y_0|\longrightarrow0.
$$
This proves <convergence in L1> and hence <convergence in probability> of the difference. If one copy starts in a <stationary distribution>, <Slutsky's theorem> shows that the other copy converges to that distribution. The argument compares paths before taking their one-time <probability distributions>; it does not require the paths ever to become exactly equal.
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