A new class of probability limit theorems

Limit Theorems in Free Probability Theory. Ii

Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line R+ and on the unit circle T we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory.

The Fundamental Limit Theorems in Probability

1. Introduction. The main purpose of this address is to explain the mathematical content and meaning of the two most important limit theorems in the modern theory of probability: the central limit theorem 1 and the recently discovered precise form of what was generally known as "KolmogorofFs celebrated law of the iterated logarithm. v The former traces its origin to the very beginnings of the t...

Class of Limit Theorems for Singular Diffusions

Limit theorems of probability theory : Sequences of independent random variables

The simple notion of statistical independence lies at the core of much that is important in probability theory. First there was the classical central limit theorem (CLT) of De Moivre and Laplace which, in its final form due to Paul Lévy, says that the sum Sn of n independent and identically distributed (i.i.d.) random variables Xj (1 ≤ j ≤ n) having finite second moments is asymptotically Gauss...

Functional Limit Theorems for A New Class of Non-Stationary Shot Noise Processes

We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of...