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Critical fluctuations of sums of weakly dependent random vectors. (English) Zbl 0792.60026

Let \(S_ n\) be sums of i.i.d. random vectors taking values in a Banach space and \(F\) be a smooth function. We study the fluctuations of \(S_ n\) under the transformed measure \(P_ n\) given by \(dP_ n/dP=\exp (nF(S_ n/n))/Z_ n\). If degeneracy occurs, then the projection of \(S_ n\) onto the degenerate subspace, properly centered and scaled, converges to a non-Gaussian probability measure with the degenerate subspace as its support. The projection of \(S_ n\) onto the non-degenerate subspace, scaled with the usual order \(\sqrt n\), converges to a Gaussian probability measure with the non-degenerate subspace as its support. The two projective limits are in general dependent. We apply this theory to the critical mean field Heisenberg model and prove a central limit type theorem for the empirical measure of this model.
Reviewer: K.Wang (Zürich)

MSC:

60F10 Large deviations
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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