
We say that the sequence X_{n} converges towards X in distribution, if
Essentially, this means that the probability that the value of X is in a given range is very similar to the probability that the value of X_{n} is in that range, if only n is large enough. This notion of convergence is used in the central limit theorems.
Convergence in distribution is also called convergence in law, since the word "law" is sometimes used as a synonym of "probability distribution." Another name is weak convergence.
We say that the sequence X_{n} converges towards X in probability if
This means that if you pick a tolerance ε and choose n large enough, then the value of X_{n} will be almost guaranteed to be within that tolerance of the value of X. This notion of convergence is used in the weak law of large numbers.
Convergence in probability implies convergence in distribution.
We say that the sequence X_{n} converges almost surely or almost everywhere or with probability 1 or strongly towards X if
This means that you are virtually guaranteed that the values of X_{n} approach the value of X. This notion of convergence is used in the strong law of large numbers.
Almost sure convergence implies convergence in probability.
We say that the sequence X_{n} converges towards X in mean or in the L^{1} norm if
This means that the expected difference between X_{n} and X gets as small as desired if n is chosen big enough. This convergence is considered in L^{p} spaces (where p = 1).
Convergence in the mean implies convergence in probability. There is no general relation between convergence in mean and almost sure convergence however.
We say that the sequence X_{n} converges towards X in mean square or in the L^{2} norm if
This means that the expected squared difference between X_{n} and X gets as small as desired if n is chosen big enough. This convergence is considered in L^{p} spaces (where p = 2).
Convergence in mean square implies convergence in mean.
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