WEBVTT
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We will now go through two
examples of convergence in
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probability.
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Our first example is
quite trivial.
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We're dealing with a sequence
of random variables Yn that
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are discrete.
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Most of the probability
is concentrated at 0.
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But there is also a small
probability of a large value.
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Because the bulk of the
probability mass is
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concentrated at 0, it is a good
guess that this sequence
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converges to 0.
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Do we have, indeed,
convergence in
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probability to 0?
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We need to check
the definition.
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So we fix some epsilon, which
is a positive number.
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And we look at the probability
of the event that our random
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variable is epsilon or more away
than what we think is the
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limit of that sequence.
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We look at that probability.
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And in this example, it is equal
to 1 over n, which goes
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to 0 as n goes to infinity.
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And this verifies that, indeed,
in this example, Yn
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converges to 0, as n goes to
infinity in probability.
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Now, we make the following
observation.
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If we are to calculate the
expected value of this random
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variable, what we get
is the following.
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We get a value of 0 with this
probability, no contribution
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to the expectation.
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But we also get a value
of n squared with
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probability 1 over n.
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And so the expected value is
equal to n, which, actually,
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goes to infinity, as
n goes to infinity.
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So we have a situation where
the sequence of the random
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variables converges to 0.
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But the expectation does
not converge to 0.
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In fact, it goes to infinity.
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And this example serves
to make the point that
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convergence in probability does
not imply convergence of
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expectations.
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The reason is that convergence
in probability has to do with
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the bulk of the distribution.
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It only cares that the tail of
the distribution has small
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probability.
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On the other hand, the
expectation is highly
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sensitive to the tail
of the distribution.
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It might be that the tail only
has a small probability.
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But if that probability is
assigned to a very large
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value, then the expectation will
be strongly affected and
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can be quite different
from the limit
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of the random variable.
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Our second example is going to
be less trivial and more
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interesting.
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Consider random variables
that are independent and
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identically distributed and
whose common distribution is
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uniform on the unit interval,
so that the
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PDF takes this form.
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Are these random variables
convergent to something?
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The answer is no.
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And the reason is that as i
increases, the distribution
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does not change.
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And it does not to
get concentrated
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around a certain number.
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The distribution remains spread
out over the entire
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unit interval.
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But let us look now at some
related random variables.
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Let us define Yn to be the
minimum of the first n of the
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X's that we get.
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So if n is equal to 4, and we
obtain these four values, Yn
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would be equal to this value.
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Notice that if we draw more
values, then the new values
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might be above the minimum, in
which case the minimum does
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not change.
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But we might also get a value
that's below the minimum, in
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which case the minimum
moves down.
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The only thing that can happen
is that the minimum goes down.
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It cannot go up.
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And this gives us
this inequality.
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So the random variables
Yn tends to go down.
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How far down?
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If n is very large, we expect
that we will obtain some X's
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whose value happens to be very
close to 0, which means that
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Yn will go down to values that
are very close to 0.
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And this leads us to conjecture
that, perhaps, Yn
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does converge to 0.
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This is always the first step
when dealing with convergence
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in probability.
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The first step is to make an
educated guess about what the
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limit might be.
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And then we want to verify
that this is, indeed, the
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correct limit.
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To verify that, what we do is we
fix some positive epsilon.
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And we look for the probability
that the distance
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of the random variable Yn from
the conjectured limit has a
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magnitude that's larger than
or equal to epsilon.
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And what we need to show is that
this quantity converges
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to 0 as n goes to infinity,
no matter what epsilon is.
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Now, because Yn is a
non-negative random variable,
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this is the same as the
probability that Yn is larger
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than or equal to epsilon.
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Now, let us distinguish
between two cases.
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If epsilon is bigger than
1, we're asking for the
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probability that Yn is larger
than or equal to a certain
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number epsilon that's
out there.
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But this probability is 0.
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There's no way that the minimum
of these uniforms will
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take a value that's larger
than some epsilon that's
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larger than 1.
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So in that case, this quantity
is equal to zero.
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And we are done.
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But we need to check that this
quantity becomes small no
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matter what epsilon is.
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So now, let us consider taking
a small epsilon that is some
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number that's less than
or equal to 1.
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For that case, let us continue
with the calculation.
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The minimum is going to be at
least epsilon, if, and only
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if, all of the random variables
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are at least epsilon.
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So this is an equivalent way
of writing this particular
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event here.
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Now, because of independence,
this is the product of the
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probabilities that each one of
the random variables is larger
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than or equal to epsilon.
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The probability that X1 is
larger than or equal to
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epsilon can be found
as follows.
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If we have here epsilon, the
probability of being larger
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than or equal to epsilon
is the probability
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of this event here.
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So it's the area of
this rectangle.
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The base of that rectangle
is 1 minus epsilon.
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And so we obtain 1 minus epsilon
for this first term.
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But because the Xi's are
identically distributed, all
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the other terms that we multiply
are also the same.
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And so the answer is this
expression here.
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Now, epsilon is a
positive number.
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So 1 minus epsilon is strictly
less than 1.
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And when we take higher powers
of a number that's less than
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1, we obtain something
that converges to 0
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as n goes to infinity.
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And that's what we
needed to verify.
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Since this is the case for any
epsilon, we conclude that the
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random variables Yn converge to
zero in the sense that we
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have defined, in probability.
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Generalizing from this example,
when we want to show
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convergence in probability, the
first step is to make a
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guess as to what is
the value that the
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sequence converges to.
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In this example, that value
was equal to 0.
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Once we have made that
conjecture, then we write down
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an expression for the
probability of being epsilon
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away from the conjectured
limit.
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And then we calculate that
probability either exactly, as
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in this example.
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Or we try to bound it somehow
and still show
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that it goes to 0.