Editing Convergence of random variables (section)

==Background==
"Stochastic convergence" formalizes the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle into a pattern. The pattern may for instance be
*[[Limit of a sequence|Convergence]] in the classical sense to a fixed value,  perhaps itself coming from a random event
*An increasing similarity of outcomes to what a purely deterministic function would produce
*An increasing preference towards a certain outcome
*An increasing "aversion" against straying far away from a certain outcome
*That the probability distribution describing the next outcome may grow increasingly similar to a certain distribution

Some less obvious, more theoretical patterns could be
*That the series formed by calculating the [[expected value]] of the outcome's distance from a particular value may converge to 0
*That the variance of the [[random variable]] describing the next event grows smaller and smaller.
These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied.

While the above discussion has related to the convergence of a single series to a limiting value, the notion of the convergence of two series towards each other is also important, but this is easily handled by studying the sequence defined as either the difference or the ratio of the two series.

For example, if the average of ''n'' [[Independence (probability theory)|independent]] random variables <math>Y_i, \ i = 1,\dots,n</math>, all having the same finite [[mean]] and [[variance]], is given by

:<math>X_n = \frac{1}{n}\sum_{i=1}^n Y_i\,,</math>

then as <math> n </math> tends to infinity, <math> X_n </math> converges ''in probability'' (see below) to the common [[mean]], <math> \mu </math>, of the random variables <math> Y_i </math>.  This result is known as the [[weak law of large numbers]].  Other forms of convergence are important in other useful theorems, including the [[central limit theorem]].

Throughout the following, we assume that <math> (X_n) </math> is a sequence of random variables, and <math> X </math> is a random variable, and all of them are defined on the same [[probability space]]  <math>(\Omega, \mathcal{F}, \mathbb{P} )</math>.