An Almost Sure Central Limit Theorem for the Hopfield Model#
نویسنده
چکیده
We prove a central limit theorem for the nite dimensional marginals of the Gibbs distribution of the macroscopicòverlap'-parameters in the Hoppeld model in the case where the number of random`patterns', M, as a function of the system size N satisses lim N"1 M(N)=N = 0, without any assumptions on the speed of convergence. The covariance matrix of the limiting gaussian distributions is diagonal and independent of the disorder for almost all realizations of the patterns.
منابع مشابه
A Central Limit Theorem for the Overlap in the Hopfield
We consider the Hopfield model with n neurons and an increasing number p = p(n) of randomly chosen patterns. Under the condition (p3 log p)/n → 0, we prove for every fixed choice of overlap parameters a central limit theorem as n → ∞, which holds for almost all realizations of the random patterns. In the special case where the temperature is above the critical one and there is no external magne...
متن کاملA moment approach for the almost sure central limit theorem for martingales
We prove the almost sure central limit theorem for martingales via an original approach which uses the Carleman moment theorem together with the convergence of moments of martingales. Several statistical applications to autoregressive and branching processes are also provided.
متن کاملThe Almost Sure Local Central Limit Theorem for the Negatively Associated Sequences
In this paper, the almost sure central limit theorem is established for sequences of negatively associated random variables: lim n→∞ (1/ log n)∑n k=1 (I(a k ≤ S k < b k )/k)P(a k ≤ S k < b k ) = 1, almost surely. This is the local almost sure central limit theorem for negatively associated sequences similar to results by Csáki et al. (1993). The results extend those on almost sure local central...
متن کاملA Note for Extension of Almost Sure Central Limit Theory
for any continuity point x of H . Several papers have dealt with logarithmic limit theorems of this kind and the above relation has been extended in various directions. Fahrner and Stadtmüller [5] gave an almost sure version of a maximum limit theorem. Berkes and Horváth [2] obtained a strong approximation for the logarithmic average of sample extremes. Berkes and Csáki [1] showed that not only...
متن کاملThe Almost Sure Local Central Limit Theorem for the Product of Partial Sums
We derive under some regular conditions an almost sure local central limit theorem for the product of partial sums of a sequence of independent identically distributed positive random variables.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1997