We are creating the world's most trusted encyclopedia and knowledge base.
Once you join us and log in, you'll be able to edit this page instantly!

Almost sure convergence

From Citizendium, the Citizens' Compendium

Jump to: navigation, search

Main Article

Talk

Definition

In this section, a formal definition of almost sure convergence will be given for complex vector-valued random variables, but it should be noted that a more general definition can also be given for random variables that take on values on more abstract topological spaces. To this end, let $(\Omega,\mathcal{F},P)$ be a probability space (in particular, $(\Omega,\mathcal{F}$ ) is a measurable space). A ( $\mathbb{C}^n$ -valued) random variable is defined to be any measurable function $X:(\Omega,\mathcal{F})\rightarrow (\mathbb{C}^n,\mathcal{B}(\mathbb{C}^n))$ , where $\mathcal{B}(\mathbb{C}^n)$ is the sigma algebra of Borel sets of $\mathbb{C}^n$ . A formal definition of almost sure convergence can be stated as follows:

A sequence $X_1,X_2,\ldots,X_n,\ldots$ of random variables is said to converge almost surely to a random variable $Y$ if $\mathop{\lim}_{k \rightarrow \infty}X_k(\omega)=Y(\omega)$ for all $\omega \in \Lambda$ , where $\Lambda \subset \Omega$ is some measurable set satisfying $P (Λ) = 1$ . An equivalent definition is that the sequence $X_1,X_2,\ldots,X_n,\ldots$ converges almost surely to $Y$ if $\mathop{\lim}_{k \rightarrow \infty}X_k(\omega)=Y(\omega)$ for all $\omega \in \Omega \backslash \Lambda'$ , where $Λ'$ is some measurable set with $P (Λ') = 0$ . This convergence is often expressed as:

$\mathop{\lim}_{k \rightarrow \infty} X_k = Y \,\,P{\rm -a.s},$

$\mathop{\lim}_{k \rightarrow \infty} X_k = Y\,\,{\rm a.s}$ .

Important cases of almost sure convergence

If we flip a coin n times and record the percentage of times it comes up heads, the result will almost surely approach 50% as $\scriptstyle n \rightarrow \infty$ .