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)


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


[1] Bolthausen, E.: Laplace approximations for sums of independent, random vectors, Probab. Theory Relat. Fields72, 305-318 (1986); Part II. Degenerate maxima and manifolds of maxima. Probab. Theory Relat. Fields,76, 167-206 (1987) · Zbl 0572.60007
[2] Comets, F., Eisele, Th.: Asymptotic dynamics, non-critical and critical fluctuations for a geometric long-range interacting model. Commun. Math. Phys.118, 531-567 (1988) · Zbl 0647.60106
[3] Ellis, R.S., Newman, C.M.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrscheinlichkeitstheor. Verw. Geb.44, 117-139 (1978) · Zbl 0364.60120
[4] Ellis, R.S., Wang, K.: Limit theorems for the empirical vector of the Curie-Weiss-Potts model, Stochastic Process. Appl.35, 59-79 (1990) · Zbl 0705.60027
[5] Hoffmann-Jørgensen, J., Pisier, G.: The law of large numbers and the central limit theorem in Banach spaces. Ann. Probab.4, 587-599 (1976) · Zbl 0368.60022
[6] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. I. Sequence Spaces, Berlin Heidelberg New York: Springer 1977; II. Function Spaces. Berlin Heidelberg New York: Springer 1979 · Zbl 0362.46013
[7] Messer, J., Spohn, H.: Statistical mechanics of the isothermal Lane-Emden equation. J. Stat. Phys.29, 561-578 (1982)
[8] Papangelou, F.: On the Gaussian fluctuations of the critical Curie-Weiss model in statistical mechanics. Probab. Theory Relat. Fields83, 265-278 (1989) · Zbl 0684.60080
[9] Papangelou, F.: Large deviations and the internal fluctuations of critical mean field systems. Stochastic Process. Appl.36, 1-14 (1990) · Zbl 0703.60023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.