# 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.

## 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.

These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied.

While the above discussion has related to the convergence of a single series to a limiting value, the notion of the convergence of two series towards each other is also important, but this is easily handled by studying the sequence defined as either the difference or the ratio of the two series.

For example, if the average of n independent random variables Yi, i = 1, ..., n, all having the same finite mean and variance, is given by

$n}={\frac {1}{n}}\sum _{{i=1}}^{n}Y_{i$

then as n tends to infinity, Template:Mvar 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 $\mathcal {F}$ .

## Convergence in distribution

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.

Convergence in distribution is the weakest form of convergence, since it is implied by all other types of convergence mentioned in this article. However convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem.

### Definition

A sequence Template:Math of random variables is said to converge in distribution, or converge weakly, or converge in law to a random variable Template:Mvar if

$n\to\infty$

for every number Template:Math at which Template:Mvar is continuous. Here Template:Mvar and Template:Mvar are the cumulative distribution functions of random variables Template:Mvar and Template:Mvar, respectively.

The requirement that only the continuity points of Template:Mvar should be considered is essential. For example if Template:Mvar are distributed uniformly on intervals Template:Math, then this sequence converges in distribution to a degenerate random variable Template:Math. Indeed, Template:Math for all n when Template:Math, and Template:Math for all Template:Math when Template:Math. However, for this limiting random variable Template:Math, even though Template:Math for all Template:Mvar. Thus the convergence of cdfs fails at the point Template:Math where Template:Mvar is discontinuous.

Convergence in distribution may be denoted as

\begin{aligned}&X_{n}\ {\xrightarrow {d}}\ X,\ \ X_{n}\ {\xrightarrow {{\mathcal {D}}}}\ X,\ \ X_{n}\ {\xrightarrow {{\mathcal {L}}}}\ X,\ \ X_{n}\ {\xrightarrow {d}}\ {\mathcal {L}}_{X},\\&X_{n}\rightsquigarrow X,\ \ X_{n}\Rightarrow X,\ \ {\mathcal {L}}(X_{n})\to {\mathcal {L}}(X),\\\end{aligned}

where $\mathcal {L}}_{X$ is the law (probability distribution) of Template:Mvar. For example if Template:Mvar is standard normal we can write $n}\,{\xrightarrow {d}}\,{\mathcal {N}$ .

For random vectors Template:Math the convergence in distribution is defined similarly. We say that this sequence converges in distribution to a random Template:Mvar-vector Template:Mvar if

$n\to\infty} \operatorname{Pr}(X_n\in A) = \operatorname{Pr$

for every Template:Math which is a continuity set of Template:Mvar.

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. This is the “weak convergence of laws without laws being defined” — except asymptotically.

In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements Template:Math converges weakly to Template:Mvar (denoted as Template:Math) if

$E}^*h(X_n) \to \operatorname{E$

for all continuous bounded functions Template:Mvar. Here E* denotes the outer expectation, that is the expectation of a “smallest measurable function Template:Mvar that dominates Template:Math”.

## Convergence in probability

The basic idea behind this type of convergence is that the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses.

The concept of convergence in probability is used very often in statistics. For example, an estimator is called consistent if it converges in probability to the quantity being estimated. Convergence in probability is also the type of convergence established by the weak law of large numbers.

### Definition

A sequence {Xn} of random variables converges in probability towards the random variable X if for all ε > 0

$n\to\infty$

Formally, pick any Template:Math and any Template:Math. Let Template:Mvar be the probability that Template:Mvar is outside the ball of radius ε centered at X. Then for Template:Mvar to converge in probability to X there should exist a number N (which will depend on ε and δ) such that for all Template:Math, Template:Mvar.

Convergence in probability is denoted by adding the letter p over an arrow indicating convergence, or using the “plim” probability limit operator:

$p}\ X,\ \ X_n \ \xrightarrow{P}\ X,\ \ \underset{n\to\infty}{\operatorname{plim}$

For random elements {Xn} on a separable metric space Template:Math, convergence in probability is defined similarly by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \forall\varepsilon>0, \Pr\big(d(X_n,X)\geq\varepsilon\big) \to 0.

### Properties

• Convergence in probability implies convergence in distribution.[proof]
• In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable X is a constant.[proof]
• Convergence in probability does not imply almost sure convergence.[proof]
• The continuous mapping theorem states that for every continuous function g(·), if $n}{\xrightarrow {p}$ , then also  $n}){\xrightarrow {p}$ .
• Convergence in probability defines a topology on the space of random variables over a fixed probability space. This topology is metrizable by the Ky Fan metric:
$\varepsilon>0:\ \Pr\big(|X-Y|>\varepsilon\big)\leq\varepsilon\big\$

or

$\mathbb E$ .

## Almost sure convergence

This is the type of stochastic convergence that is most similar to pointwise convergence known from elementary real analysis.

### Definition

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

$Pr}\!\left( \lim_{n\to\infty$

This means that the values of Template:Mvar approach the value of X, in the sense (see almost surely) that events for which Template:Mvar does not converge to X have probability 0. Using the probability space $F}, \operatorname{Pr$ and the concept of the random variable as a function from Ω to R, this is equivalent to the statement

$Pr}\Big( \omega \in \Omega : \lim_{n \to \infty$

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

$Pr}\Big( \liminf_{n\to\infty} \big\{\omega \in \Omega : | X_n(\omega) - X(\omega) | < \varepsilon \big\} \Big) = 1 \quad\text{for all$

Almost sure convergence is often denoted by adding the letters a.s. over an arrow indicating convergence:

$\mathrm{a.s.}$

For generic random elements {Xn} on a metric space (S, d), convergence almost surely is defined similarly:

$Pr}\Big( \omega\in\Omega:\, d\big(X_n(\omega),X(\omega)\big)\,\underset{n\to\infty}{\longrightarrow$

### Properties

• Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. It is the notion of convergence used in the strong law of large numbers.
• The concept of almost sure convergence does not come from a topology on the space of random variables. This means there is no topology on the space of random variables such that the almost surely convergent sequences are exactly the converging sequences with respect to that topology. In particular, there is no metric of almost sure convergence.

## Sure convergence

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

$n\to\infty$

where Ω is the sample space of the underlying probability space over which the random variables are defined.

This is the notion of pointwise convergence of sequence functions extended to sequence of random variables. (Note that random variables themselves are functions).

$\big \{}\omega \in \Omega \,|\,\lim _{{n\to \infty }}X_{n}(\omega )=X(\omega ){\big \}$

Sure convergence of a random variable implies all the other kinds of convergence stated above, but there is no payoff in probability theory by using sure convergence compared to using almost sure convergence. The difference between the two only exists on sets with probability zero. This is why the concept of sure convergence of random variables is very rarely used.

## Convergence in mean

Given a real number Template:Math, we say that the sequence Template:Mvar converges in the r-th mean (or in the Lr-norm) towards the random variable X, if the Template:Mvar-th absolute moments E(|Xn|r) and E(|X|r) of Template:Mvar and X exist, and

$n\to\infty} \operatorname{E$

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

This type of convergence is often denoted by adding the letter Lr over an arrow indicating convergence:

$n}\,{\xrightarrow {L^{r}}$

The most important cases of convergence in r-th mean are:

Convergence in the r-th mean, for r ≥ 1, implies convergence in probability (by Markov's inequality). Furthermore, if r > s ≥ 1, convergence in r-th mean implies convergence in s-th mean. Hence, convergence in mean square implies convergence in mean.

It is also worth noticing that if $L^r$ , then

$n \to \infty$

## Properties

Provided the probability space is complete:

• If $n}\ {\xrightarrow {p}$ and $n}\ {\xrightarrow {p}$ , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): X=Y almost surely.
• If $n}\ {\xrightarrow {a.s.}$ and $n}\ {\xrightarrow {a.s.}$ , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): X=Y almost surely.
• If $n}\ {\xrightarrow {L^{r}}$ and $n}\ {\xrightarrow {L^{r}}$ , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): X=Y almost surely.
• If $n}\ {\xrightarrow {p}$ and $n}\ {\xrightarrow {p}$ , then $n}+bY_{n}\ {\xrightarrow {p}$ (for any real numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): a and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): b ) and $n}Y_{n}{\xrightarrow {p}$ .
• If $n}\ {\xrightarrow {a.s.}$ and $n}\ {\xrightarrow {a.s.}$ , then $n}+bY_{n}\ {\xrightarrow {a.s.}$ (for any real numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): a and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): b ) and $n}Y_{n}{\xrightarrow {a.s.}$ .
• If $n}\ {\xrightarrow {L^{r}}$ and $n}\ {\xrightarrow {L^{r}}$ , then $n}+bY_{n}\ {\xrightarrow {L^{r}}$ (for any real numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): a and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): b ).
• None of the above statements are true for convergence in distribution.

The chain of implications between the various notions of convergence are noted in their respective sections. They are, using the arrow notation:

$\begin{matrix}{\xrightarrow {L^{s}}}&{\underset {s>r\geq 1}{\Rightarrow }}&{\xrightarrow {L^{r}}}&&\\&&\Downarrow &&\\{\xrightarrow {a.s.}}&\Rightarrow &{\xrightarrow {\ p\ }}&\Rightarrow &{\xrightarrow {\ d\ }}\end{matrix}$

These properties, together with a number of other special cases, are summarized in the following list:

$n}\ {\xrightarrow {as}}\ X\quad \Rightarrow \quad X_{n}\ {\xrightarrow {p}$
• Template:Anchor Convergence in probability implies there exists a sub-sequence $n$ which almost surely converges:
$p}\ X \quad\Rightarrow\quad X_{k_n}\ \xrightarrow{as$
$p}\ X \quad\Rightarrow\quad X_n\ \xrightarrow{d$
• Template:Anchor Convergence in r-th order mean implies convergence in probability:
$L^r}\ X \quad\Rightarrow\quad X_n\ \xrightarrow{p$
• Template:Anchor Convergence in r-th order mean implies convergence in lower order mean, assuming that both orders are greater than or equal to one:
$L^r}\ X \quad\Rightarrow\quad X_n\ \xrightarrow{L^s$ provided rs ≥ 1.
$d}\ c \quad\Rightarrow\quad X_n\ \xrightarrow{p$ provided c is a constant.
$d}\ X,\ \ |X_n-Y_n|\ \xrightarrow{p}\ 0\ \quad\Rightarrow\quad Y_n\ \xrightarrow{d$
• Template:Anchor If Template:Mvar converges in distribution to X and Yn converges in distribution to a constant c, then the joint vector (XnYn) converges in distribution to (X, c):[proof]
$d}\ X,\ \ Y_n\ \xrightarrow{d}\ c\ \quad\Rightarrow\quad (X_n,Y_n)\ \xrightarrow{d$ provided c is a constant.
Note that the condition that Template:Mvar 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 (XnYn) converges to (X, Y).
• Template:Anchor If Xn converges in probability to X and Yn converges in probability to Y, then the joint vector (XnYn) converges in probability to (XY):[proof]
$n}\ {\xrightarrow {p}}\ X,\ \ Y_{n}\ {\xrightarrow {p}}\ Y\ \quad \Rightarrow \quad (X_{n},Y_{n})\ {\xrightarrow {p}$
• Almost sure representation. Usually, convergence in distribution does not imply convergence almost surely. However for a given sequence {Xn} which converges in distribution to X0 it is always possible to find a new probability space (Ω, F, P) and random variables {Yn, n = 0, 1, ...} defined on it such that Yn is equal in distribution to Template:Mvar for each Template:Math, and Yn converges to Y0 almost surely.
• If for all ε > 0,
$n}{\mathbb {P}}\left(|X_{n$
then we say that Template:Mvar converges almost completely, or almost in probability towards X. When Template:Mvar converges almost completely towards X then it also converges almost surely to X. In other words, if Template:Mvar converges in probability to X sufficiently quickly (i.e. the above sequence of tail probabilities is summable for all Template:Math), then Template:Mvar also converges almost surely to X. This is a direct implication from the Borel-Cantelli lemma.
• If Template:Mvar is a sum of n real independent random variables:
$n}=X_{1}+\cdots +X_{n$
then Template:Mvar converges almost surely if and only if Template:Mvar converges in probability.
$matrix} X_n\xrightarrow{a.s.} X \\ |X_n| < Y \\ \mathrm{E}(Y) < \infty \end{matrix}\right\} \quad\Rightarrow \quad X_n\xrightarrow{L^1$
• A necessary and sufficient condition for L1 convergence is $n}{\xrightarrow {P}$ and the sequence (Xn) is uniformly integrable.