Almost sure convergence

From Knowino
Jump to: navigation, search

Almost sure convergence is one of the four main modes of stochastic convergence. It may be viewed as a notion of convergence for random variables that is similar to, but not the same as, the notion of pointwise convergence for real functions.

[edit] 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}.

[edit] 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

This is an example of the strong law of large numbers.

Information.svg Some content on this page may previously have appeared on Citizendium.
Personal tools