Central Limit Theorem for Random Fields

Central and Local Limit Theories in Markov Dependent Random Variables

Central Limit Theorem in Multitype Branching Random Walk

A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.

The Local Limit Theorem: A Historical Perspective

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

A Central Limit Theorem for Random Fields

A central limit theorem is proved for α-mixing random fields. The sets of locations where the random field is observed become more and more dense in an increasing sequence of domains. The central limit theorem concerns these observations. The limit theorem is applied to obtain asymptotic normality of kernel type density estimators. It turns out that in our setting the covariance structure of th...

The Limiting Behaviors of Linear Random Fields Generated by Lnqd Random Variables on Z2

In this paper we establish the central limit theorem and the strong law of large numbers for linear random fields generated by identically distributed linear negative quadrant dependent random variables on Z.

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...