WEBVTT
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We will now take a step towards
abstraction, and
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discuss the issue of
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convergence of random variables.
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Let us look at the weak
law of large numbers.
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It tells us that with high
probability, the sample mean
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falls close to the true mean
as n goes to infinity.
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We would like to interpret this
statement by saying that
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the sample mean converges
to the true mean.
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However, before we can make such
a statement, we should
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first define carefully the word
"converges." And we need
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a notion of convergence that
refers to convergence of
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random variables.
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Here's a definition.
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Suppose that we have a sequence
of random variables
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that are not necessarily
independent.
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We say that this sequence of
random variables converges in
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probability--
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that's a particular notion of
convergence we're defining.
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It converges to a certain
number if the
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following is true--
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no matter what epsilon is, as
long as it is a positive
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number, the probability that the
random variable falls far
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from this number--
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that is, epsilon or further
away from that number--
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that probability converges
to 0 as n increases.
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That is, as n increases, there
is higher and higher
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probability that Yn will
be close to this
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particular number a.
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This is the notion
of convergence
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that we have defined.
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And notice that this notion
of convergence corresponds
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exactly to what is happening
in the weak
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law of large numbers.
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And so in particular, we can
describe the weak law of large
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numbers as saying that Mn, the
sample mean, converges to mu
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as n goes to infinity, but
in a particular sense--
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in the sense of convergence
in probability.
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Let us now try to understand a
little better what convergence
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in probability really
amounts to.
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And we will do that by making a
comparison with the ordinary
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notion of convergence
of real numbers.
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When we're dealing with
convergence of numbers, we
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start with a sequence of
numbers, and we are interested
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in the statement that this
sequence converges to a
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certain limit.
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What does that mean?
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What we mean is that the
elements of the sequence
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eventually--
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that is, when n is
large enough--
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will get and stay arbitrarily
close to this particular
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number a, which is the limit.
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In terms of a picture,
here is a, the limit.
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Here is n.
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We take a small band around
this number a.
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And what we require is that the
elements of the sequence
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eventually get within this
band around the number a.
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They might get outside the
band, get inside again.
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But eventually--
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that is, after some time--
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the elements of the
sequence will only
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stay inside this band.
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Now to translate this into a
more formal mathematical
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statement, which is the
mathematical definition of the
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notion of convergence, we
have the following--
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if I give you some epsilon,
epsilon could be
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a very small number.
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I form a band around a that goes
from a minus epsilon to a
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plus epsilon.
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What I want is that there exists
a certain time, n0--
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in this picture, n0
would be here--
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such that for all times after
n0, our sequence stays within
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epsilon from a.
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That is, our sequence stays
inside this band.
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Now let us move to the case of
random variables, and see what
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convergence in probability
is talking about.
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Here, instead of a sequence of
numbers, we have a sequence of
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random variables.
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And we're interested in the
meaning of the convergence of
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the sequence of random
variables to
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a particular number.
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In words, what this means is
that if I fix a certain
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epsilon, as in this picture,
then the probability that the
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random variable falls outside
this band converges to 0.
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So the picture would
be as follows.
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We have, again, our limit.
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We fix some epsilon,
which could be an
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arbitrarily small number.
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For any fixed choice of epsilon,
we take this band,
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and for any given n, we look
into the probability that our
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random variable falls
inside that band.
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So if I am to draw the
distribution of our random
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variable, a distribution might
be something like this--
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so there is a certain
probability that it falls
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outside this band.
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But when n becomes large, this
probability of falling outside
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this band becomes very small.
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So the probability of falling
outside the band becomes tiny.
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So the bulk of the
distribution--
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that is, most of the
probability--
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is concentrated inside
this band.
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And this is true, no matter
how small epsilon is.
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If I take a much narrower band
around a, I still want all of
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the probability to
be eventually
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concentrated inside that band.
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Of course, it might
take longer.
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It might take a larger value of
n, but I want that when n
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is very large, the bulk of the
distribution is, again,
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concentrated inside
this narrow band.
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So in words, convergence in
probability means that almost
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all of the probability mass of
the random variable Yn, when n
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is large, that probability mass
get concentrated within a
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narrow band around the limit
of the random variable.
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We finally point out a few
useful properties of
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convergence in probability
that parallel well-known
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properties of convergence
of sequences.
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Suppose that we have a sequence
of random variables
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that converges in probability
to a certain number a, and
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another sequence that converges
in probability to
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some other number b.
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We do not make any assumptions
about independence.
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We do not assume the Xn's are
independent of each other.
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We do not assume that the
sequence of Xn's is
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independent of Yn.
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We then have the following
properties--
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if g is a continuous function,
then the function of the
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random variables converges to
the function of the number.
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So for example, the sequence of
random variables Xn squared
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is going to converge
to a squared.
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Another fact is that the sum of
these two random variables
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converges to the sum
of their limits.
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Both of these properties are
analogous to what happens with
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ordinary convergence
of numbers.
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And they tell us that, in some
sense, convergence in
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probability is not a very
different notion.
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We will not prove those
properties at this point, but
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they're useful to know.
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However, there's an
important caveat.
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Xn might converge to a certain
number in probability.
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However, the expected value
of Xn does not necessarily
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converge to that same limit.
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So convergence of random
variables does not imply
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convergence of expectations.
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And we will be seeing an example
where this convergence
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does not take place.