Index Vectors Can Be Converging and Continuing
This lecture discusses pointwise convergence, first for sequences of random variables and then for sequences of random vectors.
Table of contents
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Pointwise convergence of a sequence of random variables
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Pointwise convergence of a sequence of random vectors
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Solved exercises
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Exercise 1
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Exercise 2
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Exercise 3
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Let be a sequence of random variables defined on a sample space
.
Let us consider a single sample point and a generic random variable
belonging to the sequence.
is a function
. However, once we fix
, the realization
associated to the sample point
is just a real number. By the same token, once we fix
, the sequence
is just 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
, to underline that the limit depends on the specific
we have fixed.
A sequence of random variables is said to be pointwise convergent if and only if the sequence is convergent for any choice of
.
Definition Let be a sequence of random variables defined on a sample space
. We say that
is pointwise convergent to a random variable
defined on
if and only if
converges to
for all
.
is called the pointwise limit of the sequence and convergence is indicated by
Roughly speaking, using pointwise convergence we somehow circumvent the problem of defining the concept of distance between random variables: by fixing , we reduce ourselves to the familiar problem of measuring distance between two real numbers, so that we can employ the usual notion of convergence of sequences of real numbers.
Example Let be a sample space with two sample points (
and
). Let
be a sequence of random variables such that a generic term
of the sequence satisfies
We need to check the convergence of the sequences
for all
, i.e. for
and for
: (1) the sequence
, whose generic term is
, is a sequence of real numbers converging to
; (2) the sequence
, whose generic term is
, is a sequence of real numbers converging to
. Therefore, the sequence of random variables
converges pointwise to the random variable
, where
is defined as follows:
The above notion of convergence generalizes to sequences of random vectors in a straightforward manner.
Let be a sequence of random vectors defined on a sample space
, where each random vector
has dimension
.
If we fix a single sample point , the sequence
is a sequence of real
vectors.
By the standard criterion for convergence, the sequence of real vectors is convergent to a vector
if
where
is the distance between a generic term of the sequence
and the limit
.
The distance between and
is defined to be equal to the Euclidean norm of their difference:
where the second subscript is used to indicate the individual components of the vectors
and
.
Thus, for a fixed , the sequence of real vectors
is convergent to a vector
if
A sequence of random vectors is said to be pointwise convergent if and only if the sequence
is convergent for any choice of
.
Definition Let be a sequence of random vectors defined on a sample space
. We say that
is pointwise convergent to a random vector
defined on
if and only if
converges to
for all
(i.e.
).
is called the pointwise limit of the sequence and convergence is indicated by
Now, denote by the sequence of the
-th components of the vectors
. It can be proved that the sequence of random vectors
is pointwise convergent if and only if all the
sequences of random variables
are pointwise convergent:
Proposition Let be a sequence of random vectors defined on a sample space
. Denote by
the sequence of random variables obtained by taking the
-th component of each random vector
. The sequence
converges pointwise to the random vector
if and only if
converges pointwise to the random variable
(the
-th component of
) for each
.
Below you can find some exercises with explained solutions.
Exercise 1
Let the sample space be:
i.e. the sample space is the set of all real numbers between
and
. Define a sequence of random variables
as follows:
Find the pointwise limit of the sequence .
Solution
Exercise 2
Suppose the sample space is as in the previous exercise:
Define a sequence of random variables as follows:
Find the pointwise limit of the sequence .
Solution
Exercise 3
Suppose the sample space is as in the previous exercises:
Define a sequence of random variables as follows:
Define a random variable as follows:
Does the sequence
converge pointwise to the random variable
?
Solution
For , the sequence of real numbers
has limit
However, for
, the sequence of real numbers
has limit
Thus, the sequence of random variables
does not converge pointwise to the random variable
, but it converges pointwise to the random variable
defined as follows:
Please cite as:
Taboga, Marco (2021). "Pointwise convergence", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/asymptotic-theory/pointwise-convergence.
Source: https://www.statlect.com/asymptotic-theory/pointwise-convergence
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