Convergence of random variables  In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution. "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  Some less obvious, more theoretical patterns could be 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" independent random variables "Y""i", "i" = 1, ..., "n", all having the same finite mean and variance, is given by  then as "n" tends to infinity, converges "in probability" (see below) to the common mean, μ, of the random variables "Y""i". 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 ("X""n") is a sequence of random variables, and "X" is a random variable, and all of them are defined on the same probability space formula_2.   | header5 = Graphic example  With this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution. Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem.