# Critical fluctuations of sums of weakly dependent random vectors

Wang, K (1994). Critical fluctuations of sums of weakly dependent random vectors. Probability Theory and Related Fields, 98(2):229-243.

## Abstract

LetS n be sums of iid random vectors taking values in a Banach space andF be a smooth function. We study the fluctuations ofS n under the transformed measureP n given byd P n/d P=exp (nF(S n/n))/Z n. If degeneracy occurs then the projection ofS 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 ofS n onto the non-degenerate subspace, scaled with the usual order √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.

## Abstract

LetS n be sums of iid random vectors taking values in a Banach space andF be a smooth function. We study the fluctuations ofS n under the transformed measureP n given byd P n/d P=exp (nF(S n/n))/Z n. If degeneracy occurs then the projection ofS 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 ofS n onto the non-degenerate subspace, scaled with the usual order √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.

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