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Convergence in distribution

This lecture discusses convergence in distribution. We deal first with convergence in distribution of sequences of random variables and then with convergence in distribution of sequences of random vectors.

Convergence in distribution of a sequence of random variables

In the lecture entitled Sequences of random variables and their convergence we explained that different concepts of convergence are based on different ways of measuring the distance between two random variables (how "close to each other" two random variables are). The concept of convergence in distribution is based on the following intuition: two random variables are "close to each other" if their distribution functions are "close to each other".

Let [eq1] be a sequence of random variables. Let us consider a generic random variable X_n belonging to the sequence. Denote by [eq2] its distribution function. [eq3] is a function [eq4]. Once we fix x, the value [eq5] associated to the point x is a real number. By the same token, once we fix x, the sequence [eq6] is a sequence of real numbers. Therefore, for a fixed x, it is very easy to assess whether the sequence [eq7] is convergent; this is done employing the usual definition of convergence of sequences of real numbers. If, for a fixed x, the sequence [eq8] is convergent, we denote its limit by [eq9] (note that the limit depends on the specific x we have fixed). A sequence of random variables [eq1] is said to be convergent in distribution if and only if the sequence [eq11] is convergent for any choice of x (except, possibly, for some "special values" of x where [eq9] is not continuous in x).

Definition Let [eq1] be a sequence of random variables. Denote by [eq3] the distribution function of X_n. We say that [eq1] is convergent in distribution (or convergent in law) if and only if there exists a distribution function [eq9] such that the sequence [eq17] converges to [eq9] for all points $xin U{211d} $ where [eq9] is continuous. If a random variable X has distribution function [eq9], then X is called the limit in distribution (or limit in law) of the sequence and convergence is indicated by[eq21]

Note that convergence in distribution only involves the distribution functions of the random variables belonging to the sequence [eq22] and that these random variables need not be defined on the same sample space. On the contrary, the modes of convergence we have discussed in previous lectures ( pointwise convergence, almost sure convergence, convergence in probability, mean-square convergence) require that all the variables in the sequence be defined on the same sample space.

Example (Maximum of uniform random variables) Let [eq1] be a sequence of IID random variables all having a uniform distribution on the interval $left[ 0,1
ight] $, i.e., the distribution function of X_n is[eq24]Define[eq25]The distribution function of $Y_{n}$ is[eq26]Thus,[eq27]Since [eq28]we have[eq29]where [eq30] is the distribution function of an exponential random variable. Therefore, the sequence [eq31] converges in law to an exponential distribution.

Convergence in distribution of a sequence of random vectors

The definition of convergence in distribution of a sequence of random vectors is almost identical; we just need to replace distribution functions in the above definition with joint distribution functions.

Definition Let [eq1] be a sequence of Kx1 random vectors. Denote by [eq3] the joint distribution function of X_n. We say that [eq1] is convergent in distribution (or convergent in law) if and only if there exists a joint distribution function [eq9] such that the sequence [eq11] converges to [eq37] for all points $xin U{211d} ^{K}$ where [eq9] is continuous. If a random vector X has joint distribution function [eq9], then X is called the limit in distribution (or limit in law) of the sequence and convergence is indicated by[eq21]

It is important to note that for other notions of stochastic convergence (in probability, almost sure and in mean-square), the convergence of each single entry of the random vector X_n is necessary and sufficient for their joint convergence, that is, for the convergence of the vector X_n as a whole. Instead, for convergence in distribution, the individual convergence of the entries of the vector is necessary but not sufficient for their joint convergence.

More details

The following section contain more details about the concept of convergence in distribution.

Proper distribution functions

Let [eq1] be a sequence of random variables and denote by [eq3] the distribution function of X_n. Suppose that we find a function [eq9] such that [eq44]for all $xin U{211d} $ where [eq9] is continuous. How do we check that [eq46] is a proper distribution function, so that we can say that the sequence [eq1] converges in distribution?

As explained in the glossary entry on distribution functions, we just need to check that [eq9] satisfies the four properties that characterize a proper distribution function, that is, [eq46] must be increasing, right-continuous and its limits at minus and plus infinity must be [eq50]and [eq51]

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let [eq1] be a sequence of random variables having distribution functions[eq53]

Find the limit in distribution (if it exists) of the sequence [eq22].

Solution

If [eq55], then[eq56]If $1<xleq 2$, then[eq57]We now need to verify that the function[eq58]is a proper distribution function. The function is increasing, continuous, its limit at minus infinity is 0 and its limit at plus infinity is 1, hence it satisfies the four properties that a proper distribution function needs to satisfy. This implies that [eq1] converges in distribution to a random variable X having distribution function [eq46].

Exercise 2

Let [eq1] be a sequence of random variables having distribution functions[eq62]

Find the limit in distribution (if it exists) of the sequence [eq22].

Solution

If $x=0$, then[eq64]If [eq55], then[eq66]Therefore, the distribution functions [eq3] converge to the function[eq68]which is not a proper distribution function, because it is not right-continuous at the point $x=0$. However, note that the function [eq69]is a proper distribution function and it is equal to [eq70] at all points except at the point $x=0$. But this is a point of discontinuity of [eq9]. As a consequence, the sequence [eq1] converges in distribution to a random variable X having distribution function [eq9].

Exercise 3

Let [eq1] be a sequence of random variables having distribution functions[eq75]

Find the limit in distribution (if it exists) of the sequence [eq22].

Solution

The distribution functions [eq3] converge to the function[eq68]This is the same limiting function found in the previous exercise. As a consequence, the sequence [eq1] converges in distribution to a random variable X having distribution function [eq80]

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