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.