On a Markov construction of couplings
Résumé
For $N\in{\mathbb{N}$, let $\pi_N$ be the law of the number of fixed points of a random permutation of $\{1, 2, ..., N\}$. Let $\mathcal{P}$ be a Poisson law of parameter 1.
A classical result shows that $\pi_N$ converges to $\mathcal{P}$ for large $N$ and indeed in total variation
$$
\left\Vert \pi_N-\mathcal{P}\right\Vert_{\mathrm{tv}}&\leq & \f{2^N}{(N+1)!}$$
This implies that $\pi_N$ and $\mathcal{P}$ can be coupled to at least this accuracy. This paper constructs such a coupling (a long open problem) using the machinery of intertwining of two Markov chains. This method shows promise for related problems of random matrix theory.
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