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Register a new userConvex geometry of finite exchangeable laws and de Finetti style representation with universal correlated corrections
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We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If $(Z_1,...,Z_N)$ is a finitely exchangeable sequence of $N$ random variables taking values in some Polish space $X$, we show that the law $mu_k$ of the first $k$ components has a representation of the form $mu_k=int_{{mathcal P}_{frac{1}{N}}(X)} F_{N,k}(lambda) , mbox{d} alpha(lambda)$ for some probability measure $alpha$ on the set of $1/N$-quantized probability measures on $X$ and certain universal polynomials $F_{N,k}$. The latter consist of a leading term $N^{k-1}! /{small prod_{j=1}^{k-1}(N! -! j), lambda^{otimes k}}$ and a finite, exponentially decaying series of correlated corrections of order $N^{-j}$ ($j=1,...,k$). The $F_{N,k}(lambda)$ are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals $lambda$. Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis-Freedman between finite and infinite exchangeable laws.
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In 1931 de Finetti proved what is known as his Dutch Book Theorem. This result implies that the finite additivity {it axiom} for the probability of the disjunction of two incompatible events becomes a {it consequence} of de Finettis logic-operational consistency notion. Working in the context of boolean algebras, we prove de Finettis theorem. The mathematical background required is little more than that which is taught in high school. As a preliminary step we prove what de Finetti called ``the Fundamental Theorem of Probability, his main contribution both to Booles probabilistic inference problem on the object of probability theory, and to its modern reformulation known as the optimization version of the probabilistic satisfiability problem. In a final section, we give a self-contained combinatorial proof of de Finettis exchangeability theorem.
A finite form of de Finettis representation theorem is established using elementary information-theoretic tools: The distribution of the first $k$ random variables in an exchangeable binary vector of length $ngeq k$ is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided.
We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding $ell_2$-operator norms. The key to our analysis is a generalisation of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centred. Some of our results appear to be new even for such Wigner band matrices.
According to the quantum de Finetti theorem, if the state of an N-partite system is invariant under permutations of the subsystems then it can be approximated by a state where almost all subsystems are identical copies of each other, provided N is sufficiently large compared to the dimension of the subsystems. The de Finetti theorem has various applications in physics and information theory, where it is for instance used to prove the security of quantum cryptographic schemes. Here, we extend de Finettis theorem, showing that the approximation also holds for infinite dimensional systems, as long as the state satisfies certain experimentally verifiable conditions. This is relevant for applications such as quantum key distribution (QKD), where it is often hard - or even impossible - to bound the dimension of the information carriers (which may be corrupted by an adversary). In particular, our result can be applied to prove the security of QKD based on weak coherent states or Gaussian states against general attacks.
We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $varepsilon$ going to zero as $N$ tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework.
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