# Almost sure convergence

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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 ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ be a probability space (in particular, ${\displaystyle (\Omega ,{\mathcal {F}}}$) is a measurable space). A (${\displaystyle \mathbb {C} ^{n}}$-valued) random variable is defined to be any measurable function ${\displaystyle X:(\Omega ,{\mathcal {F}})\rightarrow (\mathbb {C} ^{n},{\mathcal {B}}(\mathbb {C} ^{n}))}$, where ${\displaystyle {\mathcal {B}}(\mathbb {C} ^{n})}$ is the sigma algebra of Borel sets of ${\displaystyle \mathbb {C} ^{n}}$. A formal definition of almost sure convergence can be stated as follows:

A sequence ${\displaystyle X_{1},X_{2},\ldots ,X_{n},\ldots }$ of random variables is said to converge almost surely to a random variable ${\displaystyle Y}$ if ${\displaystyle \mathop {\lim } _{k\rightarrow \infty }X_{k}(\omega )=Y(\omega )}$ for all ${\displaystyle \omega \in \Lambda }$, where ${\displaystyle \Lambda \subset \Omega }$ is some measurable set satisfying ${\displaystyle P(\Lambda )=1}$. An equivalent definition is that the sequence ${\displaystyle X_{1},X_{2},\ldots ,X_{n},\ldots }$ converges almost surely to ${\displaystyle Y}$ if ${\displaystyle \mathop {\lim } _{k\rightarrow \infty }X_{k}(\omega )=Y(\omega )}$ for all ${\displaystyle \omega \in \Omega \backslash \Lambda '}$, where ${\displaystyle \Lambda '}$ is some measurable set with ${\displaystyle P(\Lambda ')=0}$. This convergence is often expressed as:

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

or

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

## 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 ${\displaystyle \scriptstyle n\rightarrow \infty }$.

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