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Convergence of random variables

In probability theory, there exist several different notions of convergence of random variables. The convergence (in one of the senses presented below) of sequences of random variables to some limiting random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. For example, if the average of n independent, identically distributed random variables Yi, i = 1, ..., n, is given by

then as n goes to infinity, Xn converges in probability (see below) to the common mean, μ, of the random variables Yi. 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 (Xn) is a sequence of random variables, and X is a random variable, and all of them are defined on the same probability space (Ω, F, P).

Table of contents
1 Convergence in distribution
2 Convergence in probability
3 Almost sure convergence
4 Convergence in rth mean
5 Converse implications
6 References

Convergence in distribution

We say that the sequence Xn converges towards X in distribution, if

for every real number a at which the cumulative distribution function of the limiting random variable X is continuous. 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 Xn 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 the weakest form of convergence (it is sometimes called weak convergence), and does not, in general, imply any other mode of convergence. However, convergence in distribution is implied by all other modes of convergence mentioned in this article, and hence, it is the most common and often the most useful form of convergence of random variables.

A useful result, which may be employed in conjunction with laws of large numbers and the central limit theorem, is that if a function  g: RR  is continuous, then if  Xn  converges in distribution to  X, then so too does  g(Xn)  converge in distribution to  g(X). (This may be proved using Skorokhod's representation theorem.)

Convergence in distribution is also called convergence in law, since the word "law" is sometimes used as a synonym of "probability distribution."

Convergence in probability

We say that the sequence Xn converges towards X in probability if

for every ε > 0. Convergence in probability is, indeed, the (pointwise) convergence of probabilities. Pick any ε > 0 and any δ > 0. Let Pn be the probability that Xn is outside a tolerance ε of X. Then, if Xn converges in probability to X then there exists a value N such that, for all nN, Pn is itself less than δ.

Convergence in probability implies convergence in distribution, and is the notion of convergence used in the weak law of large numbers.

Almost sure convergence

We say that the sequence Xn 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 Xn approach the value of X, in the sense (see almost surely) that events for which '\'Xn does not converge to X have probability 0. Using the probability space (Ω, F'', P) and the concept of the random variable as a function from Ω to R, this is equivalent to the statement

Almost sure convergence implies convergence in probability, and hence implies convergence in distribution. It is the notion of convergence used in the strong law of large numbers.

Convergence in rth mean

We say that the sequence Xn converges in rth mean or in the Lr norm towards X, if r ≥ 1, E|Xn| < ∞ for all n, and

where the operator E denotes the expected value. Convergence in rth mean tells us that the expectation of the rth power of the difference between Xn and X converges to zero.

The most important cases of convergence in rth mean are:

Convergence in rth mean, for r ≥ 1, implies convergence in probability, while if r > s ≥ 1, convergence in rth mean implies convergence in sth mean. Hence, convergence in mean square implies convergence in mean.

Converse implications

The chain of implications between the various notions of convergence, above, are noted in their respective sections, but it is sometimes important to establish converses to these implications. No other implications other than those noted above hold in general, but a number of special cases do permit converses:

then Xn converges almost surely to X. In other words, if Xn converges in probability to X sufficiently quickly (i.e. the above sum converges for all ε > 0), then Xn also converges almost surely to X.


  1. G.R. Grimmett and D.R. Stirzaker (1992). Probability and Random Processes, 2nd Edition. Clarendon Press, Oxford, pp 271--285. ISBN 0198536658.