This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution.

The concept is important 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 certain properties of 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.

This result is known as the weak law of large numbers.

is a random variable, and all of them are defined on the same probability space

Loosely, 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.

However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem.

of real-valued random variables, with cumulative distribution functions

, then this sequence converges in distribution to the degenerate random variable

We say that this sequence converges in distribution to a random k-vector X if for every

The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes.

[1] In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements {Xn} converges weakly to X (denoted as Xn ⇒ X) if for all continuous bounded functions h.[2] Here E* denotes the outer expectation, that is the expectation of a “smallest measurable function g that dominates h(Xn)”.

A sequence {Xn} of random variables converges in probability towards the random variable X if for all ε > 0 More explicitly, let Pn(ε) be the probability that Xn is outside the ball of radius ε centered at X.

Notice that for the condition to be satisfied, it is not possible that for each n the random variables X and Xn are independent (and thus convergence in probability is a condition on the joint cdf's, as opposed to convergence in distribution, which is a condition on the individual cdf's), unless X is deterministic like for the weak law of large numbers.

At the same time, the case of a deterministic X cannot, whenever the deterministic value is a discontinuity point (not isolated), be handled by convergence in distribution, where discontinuity points have to be explicitly excluded.

Convergence in probability is denoted by adding the letter p over an arrow indicating convergence, or using the "plim" probability limit operator: For random elements {Xn} on a separable metric space (S, d), convergence in probability is defined similarly by[6] Not every sequence of random variables which converges to another random variable in distribution also converges in probability to that random variable.

As an example, consider a sequence of standard normal random variables

To say that the sequence Xn converges almost surely or almost everywhere or with probability 1 or strongly towards X means that

and the concept of the random variable as a function from Ω to R, this is equivalent to the statement

Using the notion of the limit superior of a sequence of sets, almost sure convergence can also be defined as follows:

Almost sure convergence is often denoted by adding the letters a.s. over an arrow indicating convergence: For generic random elements {Xn} on a metric space

are independent, second Borel Cantelli Lemma ensures that

To say that the sequence of random variables (Xn) defined over the same probability space (i.e., a random process) converges surely or everywhere or pointwise towards X means

This is why the concept of sure convergence of random variables is very rarely used.

Given a real number r ≥ 1, we say that the sequence Xn converges in the r-th mean (or in the Lr-norm) towards the random variable X, if the r-th absolute moments

(|X|r ) of Xn and X exist, and where the operator E denotes the expected value.

Provided the probability space is complete: The chain of implications between the various notions of convergence are noted in their respective sections.

They are, using the arrow notation: These properties, together with a number of other special cases, are summarized in the following list: This article incorporates material from the Citizendium article "Stochastic convergence", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.