Convergence of transformations
This lecture discusses some well-known results on the convergence of transformed sequences of random vectors.
Continuous mapping theorem
Suppose a sequence of random vectors
![[eq1]](s.gif)
converges to a random vector
(either in probability, in distribution or almost surely). Now, take a
transformed sequence
,
where
is a function. Under what conditions is
also a convergent sequence?
The following proposition, known as Continuous mapping theorem, states that convergence is preserved by continuous transformations:
Proposition (Continuous
mapping)_
Let
be a sequence of
-dimensional
random vectors. Let
be a continuous function.
Then:![[eq6]](http://images1.statlect.com/convergence_of_transformations__9.png)
where
denotes convergence in probability,
denotes almost sure convergence and
denotes
convergence in distribution.
See e.g. Shao, J. (2003) "Mathematical statistics", Springer.
The following subsections present some important applications of the continuous mapping theorem.
Sums and products of sequences converging in probability
An important implication of the continuous mapping theorem is that arithmetic operations preserve convergence in probability:
Proposition_
If
and
.
Then:
First of all, note that convergence in
probability of
and of
implies their joint convergence in probability (see the lecture entitled
Convergence in probability), i.e. their
convergence as a vector:
![[eq15]](http://images2.statlect.com/convergence_of_transformations__18.png)
Now, the sum and the product are continuous functions of the operands. Thus,
for
example:![[eq16]](http://images1.statlect.com/convergence_of_transformations__19.png)
where
is a continuous function, and, using the continuous mapping
theorem:![[eq17]](http://images2.statlect.com/convergence_of_transformations__21.png)
where
denotes a limit in probability.
Sums and products of sequences converging almost surely
Everything that was said in the previous subsection applies, with obvious modifications, also to almost surely convergent sequences:
Proposition_
If
and
,
then:![[eq20]](http://images1.statlect.com/convergence_of_transformations__25.png)
Similar to previous proof. Just replace convergence in probability with almost sure convergence.
Sums and products of sequences converging in distribution
For convergence almost surely and convergence in probability, the convergence
of
and
individually implies their joint convergence as a vector (see the previous two
proofs), but this is not the case for convergence in distribution. Therefore,
to obtain preservation of convergence in distribution under arithmetic
operations, we need the stronger assumption of joint convergence in
distribution:
Proposition_
If
![[eq23]](http://images2.statlect.com/convergence_of_transformations__28.png)
then:![[eq24]](http://images1.statlect.com/convergence_of_transformations__29.png)
Again, similar to the proof for convergence in probability, but this time joint convergence is already in the assumptions.
Slutski's Theorem
Slutski's theorem concerns the convergence in distribution of the sum and product of two sequences, one converging in distribution and the other converging in probability to a constant:
Proposition (Slutski)_
If
and
,
where
is a constant,
then:
It is possible to prove (see e.g. A. W. van
der Vaart (2000) Asymptotic Statistics, Cambridge
University Press) that
and
imply:![[eq30]](http://images2.statlect.com/convergence_of_transformations__36.png)
but
this in turn implies convergence under arithmetic operations (see above).
More details
Convergence of ratios
As a byproduct of the propositions stated above, we also have:
Proposition_
If a sequence of random variables
converges to
,
thenprovided
is almost surely different from
(we did not specify the kind convergence, which can be can be in probability,
almost surely or in distribution).
This is a consequence of the Continuous
mapping theorem and of the fact that
is
a continuous function for
.
As a consequence:
Proposition_
If two sequences of random variables
and
converge to
and
respectively,
thenprovided
is almost surely different from
.
Convergence can be in probability, almost surely or in distribution (but the
latter requires joint convergence in distribution of
and
).
This is a consequence of the fact that the
ratio can be written as a
product![[eq39]](http://images1.statlect.com/convergence_of_transformations__53.png)
The
first operand of the product converges by assumption. The second converges
because of the previous proposition. Therefore, their product converges
because convergence is preserved under products.
Random matrices
The Continuous mapping theorem and Slutski's theorem apply also to random matrices, because random matrices are just random vectors whose elements have been arranged into columns.
In particular:
-
if two sequences or random matrices are convergent, then also the sum and the product of their terms are convergent (provided their dimensions are such that they can be summed or multiplied);
-
if a sequence of square random matrices
converges to a random matrix
,
then the sequence of inverse matrices
converges to the random matrix
(provided the matrices are invertible). This is a consequence of the fact that
matrix inversion is a continuous transformation.
Solved exercises
Below you can find some exercises with explained solutions: