# Sequences and Limit Theorems

Sequences :: Convergence :: Limit :: Central Limit

## Modes of convergence of sequences of r.v.s

Suppose is a sequence of random variables defined on . How can we define a limit of this sequence? As it turns out, there are several different (and inequivalent) ways of defining convergence.

## Almost sure convergence

This is a very strong form of convergence, and usually quite difficult
to prove.

One tool for showing a.s. convergence is the following fact:

a.s. if and only if

## Mean-square convergence

This is a strong mode of convergence which is usually easier to show
than a.s. It is widely used in engineering.

We write
(m.s.) or
(q.m.)
''quadratic mode.''

There is a Cauchy criterion for m.s. convergence: If
for all
, then
converges in
mean-square if and only if

Here is an interesting fact: If (m.s.) and (a.s.), then (a.s.).

## Convergence in Probability

## Convergence in Distribution

Note from this example that the
values don't really ''approach''
any value -- the values are still 1 and 0. This is in distinction to
the first three modes of convergence, in which
in some sense.

By the definition of this mode of convergence, we don't have to worry
about the points of discontinuity of
.

## Why and Which?

We have defined several different modes of convergence. Why so many?
The basic answer is that they are inequivalent -- one does not meet
all the analytical needs. Some are stronger than others.

So convergence in distribution is weaker than i.p., m.s. or a.s.

In general, none of the implications can be reversed. And m.s. and a.s. do not imply each other. (Venn diagram - dist. on the outside, then i.p., with m.s. and a.s. overlapping inside.)

## Some examples of invalid implications

To see which modes are ''stronger'' than others, we can consider some
counterexamples.

Some other relationships:

- If (i.p.) then there is a subsequence such that (a.s.)
- If and there is a r.v. with finite second moment such that (a.s.) for every , then (m.s.).
- If (in distribution), then (i.p.)