Convergence of random variables

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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.

Contents

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

  • 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

Some less obvious, more theoretical patterns could be

  • That the probability distribution describing the next outcome may grow increasingly similar to a certain distribution
  • 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.

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