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.

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

Definition
Let
be a sequence of random variables. Denote by
the distribution function of
.
We say that
is **convergent in distribution** (or convergent in law) if and
only if there exists a distribution function
such that the sequence
converges to
for all points
where
is continuous. If a random variable
has distribution function
,
then
is called the **limit in distribution **(or limit in law) of the
sequence and convergence is indicated
by

Note that convergence in distribution only involves the distribution functions of the random variables belonging to the sequence 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 be a sequence of IID random variables all having a uniform distribution on the interval , i.e., the distribution function of isDefineThe distribution function of isThus,Since we havewhere is the distribution function of an exponential random variable. Therefore, the sequence converges in law to an exponential distribution.

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
be a sequence of
random
vectors. Denote by
the joint distribution function of
.
We say that
is **convergent in distribution** (or convergent in law) if and
only if there exists a joint distribution function
such that the sequence
converges to
for all points
where
is continuous. If a random vector
has joint distribution function
,
then
is called the **limit in distribution **(or limit in law) of the
sequence and convergence is indicated
by

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 is necessary and sufficient for their joint convergence, that is, for the convergence of the vector 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.

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

Let be a sequence of random variables and denote by the distribution function of . Suppose that we find a function such that for all where is continuous. How do we check that is a proper distribution function, so that we can say that the sequence converges in distribution?

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

Below you can find some exercises with explained solutions.

Let be a sequence of random variables having distribution functions

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

Solution

If , thenIf , thenWe now need to verify that the functionis a proper distribution function. The function is increasing, continuous, its limit at minus infinity is and its limit at plus infinity is , hence it satisfies the four properties that a proper distribution function needs to satisfy. This implies that converges in distribution to a random variable having distribution function .

Let be a sequence of random variables having distribution functions

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

Solution

If , thenIf , thenTherefore, the distribution functions converge to the functionwhich is not a proper distribution function, because it is not right-continuous at the point . However, note that the function is a proper distribution function and it is equal to at all points except at the point . But this is a point of discontinuity of . As a consequence, the sequence converges in distribution to a random variable having distribution function .

Let be a sequence of random variables having distribution functions

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

Solution

The distribution functions converge to the functionThis is the same limiting function found in the previous exercise. As a consequence, the sequence converges in distribution to a random variable having distribution function

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