WEBVTT
Kind: captions
Language: en
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We will have another example in which it is
going to be only the - it won't be a success
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itself let me start with the example in which
this stochastic process is a strict sense
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stationary process in the given Xt is a strict
sense stationary process with finite second
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order moment. So you don't want the finite
a second order moment for the strict sense
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stationary process but I have taken as an
example the given Xt is going to be a strict
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sense stationary process along with the finite
second order moment.
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Now I am going to define the another stochastic
process with the random variable Y of t that
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is a bt plus X of t. So this is going to be
a stochastic process. This is a stochastic
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process Yt. Now we want to check whether the
Yt is going to be a stationary - strict sense
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stationary process or not as well as whether
this is going to be wide sense stationary
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or not. The Xt is a strict sense stationary
process suppose you find out the mean for
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this random variable if you find out the mean
for the Y of t where a and b are constant
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therefore this is going to be a function of
t since a and b are constant the mean of Y
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of t is a function of t therefore this is
a function of t since it is not satisfying
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the first property of the first condition
to become a wide sense stationary process
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therefore the Y of t is not a wide sense
stationary process. We started with the strict
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sense stationary process and we created a
new stochastic process Y of t that is a plus
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bt plus Xt where a and b are constant. Now
if you find out the mean of Y of t mean function
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that is going to be a function of t that is
nothing but that depends on t therefore Yt
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is not going to be a wide sense stationary
process whereas Xt is a strict sense.
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Now similarly you can cross check whether
the joint distribution of a Y of t shifted
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by h, t is shifted by h you can conclude this
is also not going to be a since it is a function
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of a t since it is the mean is going to be
a function of t and Y of t also involves the
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function of t as well as X of t even though
X of t is a strict sense stationary process
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the way you made a plus bt plus Xt you will
land up the joint distributions are going
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to be different by the t with the shifted
t plus h it won't be satisfied. Therefore
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you can conclude Y of t is not a strict sense
stationary process also. That means from this
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example we can conclude whenever you have
a strict sense stationary process if you make
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it a plus bt plus Xt definitely the Y of t
is not going to be a wide sense stationary
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process as well as a strict sense stationary
process.
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We will go for the third example. In this
third example let me start with the stochastic
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process, here this each random variables are
uncorrelated random variables with the mean
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of each random variable is going to be some
constant k which may be assume to be 0 in
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some situation. So in general you keep the
mean of each random variable is going to be
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some constant k and you make X of m X of n
that is going to be its variance for m is
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equal to n and for all other quantity you
make it 0. Not only this each random variables
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are uncorrelated random variable that means
if you find out the correlation coefficient
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that is going to be 0 and the mean is going
to be constant and expectation of the product
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of any two random variables if they are different
it is 0 and obviously if they are same since
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you make the assumption therefore this is
going to be a variance Sigma square. If you
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cross check all the properties of all the
conditions of the wide-sense stationarity
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property starting with the mean function and
second order moment exists that is finite
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and covariance function of any two random
variables is going to be a function of only
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the difference. There all those three conditions
are going to be satisfied therefore you can
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come to the conclusion, I am not working out
here. This is going to be a weakly stationary
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process or wide-sense stationary process or
it is going to be call it as a covariance
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stationary process also. And This stochastic
process is also called a white noise process.
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This is very important in the signal processing.
You keep the uncorrelated random variable
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with this assumption the mean is going to
be a constant which may be 0 and the product
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of expectation is going to be these values
and this is going to be a weakly stationary
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process in the sense it satisfies all three
conditions of that weak sense or white sense
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stationary process and this stochastic process
is called a white noise process.
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Note that this stochastic process we didn't
make the distribution of each random variable
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Xn what's the distribution of Xn is not defined
here; without that we give the all the assumptions
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of the mean and variance therefore this is
going to be very useful in the time series
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analysis as well as the signal processing
and this particular stochastic process is
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called the white noise and sometimes we make
the assumption the Xns are going to be normally
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distributed random variable also but in general
we want to define - we want to give what is
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the assumption what is the distribution of
Xn without that this stochastic process is
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going to be call it as a white noise process.
Addition to the white sense stationary process
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one can assume
that Ergodic property also satisfied along
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with white sense
stationary property.
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For illustration purpose we have discussed
Bernoulli process. That means the given stochastic
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process is a wide sense stationary process
as well as it is the Ergodic property is also
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satisfied in that case the mean function is
going to be a sum independent of t that you
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can make it as the mu and auto covariance
function is going to be a R of tau only because
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it is a wide sense stationary process therefore
the mean is independent of t and the auto
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correlation function is going to be a function
with the only tau and we have Ergodic property
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therefore you can find the mean can be estimated
from the time average. So this is possible
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only if the Ergodic property satisfied so
the mean can be estimated with the up arrow
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that means the estimator estimation of a mean
that is same as 1divided by 2 times t and
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minus t 2T of X of t dt. So this is possible
as long as the stochastic process is - so
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in general I define t belonging to T that
T is different from this T so here you have
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the time interval of length 2T within that
2T if you find out the time average and that
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time average quantity is going to be the estimation
for the mean that means if mu t converges
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in the squared mean to mu as a t tends to
infinity then the process is going to be a
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mean Ergodic stochastic process is going to
be call it as a mean Ergodic process. Similarly
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one can estimate other higher order moments
also provided the process is Ergodic with
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respect to those moments. So here I have made
the Ergodic with respect to the mean therefore
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you are estimating the mean with the Ergodic
property. Similarly if this given stochastic
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process is satisfying the Ergodic property
with the higher order moment then those measures
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also can be estimated in the same thing. So
here the mu t converges in squared mean to
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mu as t tends to infinity. So that is the
conclusion we are getting from the Ergodic
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property along with the wide sense stationary
property.
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With this let me stop today's lecture and
some more examples for the stationary process.
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So maybe those example maybe a white sense
stationary process or strict sense stationary
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process that I am going to give in the next
lecture. So today's lecture what I have covered
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is what is a stationary process and to conclude
or for a given stochastic process is going
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to be a stationary process for that we have
given few some definitions. So with those
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definitions we can come to the conclusion
the given stochastic process is going to be
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a wide sense stationary process or strict
sense stationary process and I have given
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three examples in today's lecture and I will
give two more examples of a stationary process
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in the next lecture.
Then I will go for some simple stationary
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process that is auto regressive process and
moving average process and some more stochastic
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process for the stationary process example
I will give it in the lecture 2.
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With this today's lecture is over. Thanks.