Acceptable random variables in non-commutative probability spaces

Topic 2: Probability 2.1 Probability spaces and random variables

x x · P(X = x) , where the summation is over the range of X. Here, P(X = x) = P({ω ∈ Ω : X(ω) = x}). The notation above implicitly assumes that range of X is a discrete set that we can enumerate. But we’ll often encounter a continuous random variable X whose range is a continuous space, like the real line R or some subset thereof. In the cases we’ll consider, the random variable X will have a p...

Non-commutative Rational Functions in Strongly Convergent Random Variables

Random matrices like GUE, GOE and GSE have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and Thorbørnsen. It is called strong convergence property and then more random matrices with this property are followed. In general, the definition can be stated for a sequence of tuples over some C∗-algebras. In t...

Integral Non - Commutative Spaces

Integral Non-commutative Spaces

A non-commutative space X is a Grothendieck category ModX. We say X is integral if there is an indecomposable injective X-module EX such that its endomorphism ring is a division ring and every X-module is a subquotient of a direct sum of copies of EX . A noetherian scheme is integral in this sense if and only if it is integral in the usual sense. We show that several classes of non-commutative ...

A Probability Space based on Interval Random Variables

This paper considers an extension of probability space based on interval random variables. In this approach, first, a notion of interval random variable is introduced. Then, based on a family of continuous distribution functions with interval parameters, a concept of probability space of an interval random variable is proposed. Then, the mean and variance of an interval random variable are intr...

Sample Spaces, Random Variables

1 Probabilities In talking about probabilities, the fundamental object is Ω, the sample space. Points (elements) in Ω are denoted (generically) by ω. We assign probabilities to subsets of Ω. Assume for the moment that Ω is finite or countably infinite. Thus Ω could be the space of all possible outcomes when a coin is tossed three times in a row or say, the set of positive integers. A probabilit...