Second Order Regular Variation , Convolution and the Central Limit Theorem 3

Asymptotic normality of the integrated square error of a density estimator in the convolution model

In this paper we consider a kernel estimator of a density in a convolution model and give a central limit theorem for its integrated square error (ISE). The kernel estimator is rather classical in minimax theory when the underlying density is recovered from noisy observations. The kernel is fixed and depends heavily on the distribution of the noise, supposed entirely known. The bandwidth is not...

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...

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.

Conditionally monotone independence

We define the notion of conditionally monotone product as a part of conditionally free product, which naturally includes monotone and Boolean products. Then we define conditionally monotone cumulants which are useful to calculate the limit distributions in central limit theorem and Poisson’s law of small numbers. Moreover, we introduce deformed convolutions arising from the conditionally monoto...