Almost sure convergence

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.

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 $P(\Lambda )=1$. 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 $\Lambda '$ is some measurable set with $P(\Lambda ')=0$. This convergence is often expressed as:

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

or

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

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 $\scriptstyle n\rightarrow \infty$.

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