Asymptotics of Discrete Schrödinger Bridges via Chaos Decomposition

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Consider the problem of matching two independent sets of $N$ i.i.d. observations from two densities $\rho_0$ and $\rho_1$ in $\mathbb{R}^d$. For an arbitrary continuous cost function, the optimal assignment problem looks for the matching that minimizes the total cost. We consider instead the problem where each matching is endowed with a Gibbs probability weight proportional to the exponential of the negative total cost of that matching. Viewing each matching as a joint distribution with $N$ atoms, we then take a convex combination with respect to the above Gibbs probability measure. We show that this resulting random joint distribution converges, as $N\rightarrow \infty$, to the solution of a variational problem, introduced by Föllmer, called the Schrödinger problem. Finally, we prove limiting Gaussian fluctuations for this convergence in the form of Central Limit Theorems for integrated test functions. This is enabled by a novel chaos decomposition for permutation symmetric statistics, generalizing the Hoeffding decomposition for U-statistics. Our results establish a novel passage for the transition from discrete to continuum in Schrödinger’s lazy gas experiment.