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Large deviations and the internal fluctuations of critical mean field systems. (English) Zbl 0703.60023

Consider a system of kn random variables \(X_ 1^{(kn)},...,X_{kn}^{(kn)}\) (k a fixed positive integer, \(n\to \infty)\) with joint distribution \[ P(X_ 1^{(kn)}\in dx_ 1,...,X_{kn}^{(kn)}\in dx_{kn})=c_ n\exp \{kn G((X_ 1+...+X_{kn})/kn)\}\prod^{kn}_{i=1}\rho (dx_ i), \] where \(\rho\) is a probability measure on the real line with mean 0, variance 1 and entropy function H. Define \[ S^ j_ n=X^{(kn)}_{(j- 1)n+1}+...+X_{jn}^{(kn)},\quad j=1,...,k. \] Assuming that, for some \(\lambda >0\) and \(p>2\), \[ G(x)-H(x)=-\lambda | x|^ p+o(| x|^ p)\quad as\quad x\to 0, \] the author’s main result proves that \(S^ k_ n/n^{1-1/p}\) has an asymptotic probability density \(const \exp (-k\lambda | x|^ p)\) and is asymptotically independent of the random vector \[ n^{-1/2}(S^ 1_ n-S^ k_ n,...,S_ n^{k- 1}-S^ k_ n), \] the latter converging in distribution to \((Y_ 1- Y_ k,...,Y_{k-1}-Y_ k)\), where \(Y_ 1,...,Y_ k\) are i.i.d. normal (0,1) random variables.
This establishes, for rather general G, asymptotic Gaussian distributions of the internal fluctuations of the system together with an asymptotic non-Gaussian distribution along the principal diagonal. The proofs require only some basic facts about one-dimensional large deviations, going back to Cramér’s classical work.
Reviewer: J.Steinebach

MSC:

60F10 Large deviations
60F05 Central limit and other weak theorems
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