## 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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### References:

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