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.

References

1. P. Billingsley, Probability and Measure (2 ed.), ser. Wiley Series in Probability and Mathematical Statistics, Wiley, 1986.
2. D. Williams, Probability with Martingales, Cambridge : Cambridge University Press, 1991.
3. E. Wong and B. Hajek, Stochastic Processes in Engineering Systems, New York: Springer-Verlag, 1985.

See also

• Stochastic convergence
• Convergence in distribution
• Convergence in probability
• Convergence in r-th order mean

Related topics

• Stochastic variable
• Stochastic process
• Stochastic diffential equation