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 be a probability space (in particular, ) is a measurable space). A (-valued) random variable is defined to be any measurable function , where is the sigma algebra of Borel sets of . A formal definition of almost sure convergence can be stated as follows:

A sequence of random variables is said to converge almost surely to a random variable if for all , where is some measurable set satisfying . An equivalent definition is that the sequence converges almost surely to if for all , where is some measurable set with . This convergence is often expressed as:

or

.

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 .

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