Almost sure convergence: Difference between revisions
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==References== | ==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== | ==See also== |
Revision as of 07:29, 16 October 2007
Almost sure convergence is one of the four main modes of stochastic convergence. It may be viewed as a notion of convergece 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 set satisfying . An equivalent definition is that the sequence converge almost surely to if for all , where is some 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
Related topics