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 sets. 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 Borel set 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 converge 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.
References
- D. Williams, Probability with Martingales, Cambridge : Cambridge University Press, 1991.
- P. Billingsley, Probability and Measure (2 ed.), ser. Wiley Series in Probability and Mathematical Statistics. Wiley, 1986.
- E. Wong and B. Hajek, Stochastic Processes in Engineering Systems, Springer-Verlag, New York, 1985.
See also
- Stochastic convergence
- Convergence in distribution
- Convergence in probability
- Convergence in rth order mean
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