# 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 behaviour that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behaviour can be characterised: two readily understood behaviours 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.

Read more about Convergence Of Random VariablesBackground, Convergence in Distribution, Convergence in Probability, Almost Sure Convergence, Sure Convergence, Convergence in Mean, Properties

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Convergence Of Random Variables - Properties
... The chain of implications between the various notions of convergence are noted in their respective sections ... with a number of other special cases, are summarized in the following list Almost sure convergence implies convergence in probability Convergence in probability implies there exists a sub-sequence which almost ... that Yn converges to a constant is important, if it were to converge to a random variable Y then we wouldn’t be able to conclude that (Xn, Yn) converges to (X, Y) ...

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