Now we are moving into the fourth definition that is auto correlation. Auto correlation
function. The way we have defined the covariance function. Now we are defining the auto correlation
function. It is defined with the notation R of s,t that is nothing but or we can write
it in the terms of expectation. Expectation of X of t, X of s minus expectation of X of
t into expectation of X of s divided by the square root of variance of X of t and the
square root of variance of X of s. So the numerator can be written covariance of X of
a t,s divided by square root of variance of X of t square root of variance of X of s.
So this is going to be used with the notation R of s,t and this is going to be auto correlation
function for the random variable X of t and X of s.
So it's basically describes the correlation between values of process at the different
time points s and t. Sometimes we assume the - we assume R of s,t depends only on absolute
of t minus s. In the later case when you are discussing the stationary process it is going
to be depends only on the interval length not the actual time. Therefore the R of s,t
is going to be depend only on the length that t minus s in absolute not the actual s and
t. Therefore by assuming R of s,t is going to be a only depends on t minus s we can have
R of instead of for two variables I can use the one variable as the R of tau that is nothing
but the expectation of X of t minus mu multiplied by X of t plus tau minus mu the expectation
of that product divided by Sigma square. So here I have made the one more assumption
the m of t that is nothing but the expectation of X of t that is going to be mu and the variance
of X of t is going to be Sigma square. With that assumption only the R of tau is going
to be expectation of this product divided by Sigma square where the variance of X of
t is going to be not a function of t it is a constant that is Sigma square and similarly
the mean function expectation of X of t is going to be mu that is also independent of
t therefore I can simplify this R of s,t the product expectation minus individual expectation
that can be simplified as expectation of this product.
So basically this is evaluated at X of t and X of t plus tau and that difference is going
to be tau and this is also going to be even function that means it has R of tau is same
as R of minus tau and this auto correlation function is used in time series analysis as
well as a signal processing. In the signal processing we assume that the signal the corresponding
time series satisfying the stationary property therefore the stationarity property implies
the auto correlation function is going to be depends only on the length of the interval
not the actual time. Therefore, this R of tau will be used in the signal processing
as well as in general time series analysis also.
The fifth definition so we are covering the different definitions which we want the fifth
definition first we started with the mean function. Second we started with the second
order stochastic process. Then third we start - third we have given the covariance function
and the fourth we have given the auto correlation function. Now we are giving the definition
that is independent increments. If for every t1 less than t2 less than tn
the random variables X of t2 minus X of t1, X of t3 minus X of t2 so on till X of tn minus
X of tn minus 1 are mutually independent random variables for all n then we say the corresponding
stochastic process is having independent increment property.
So whenever you take a few t's t1,t2, tn and the increments that is X of t2 minus X of
t1 like that till X of tn minus Xtn minus 1 so these are all going to be the increment
and each one is random variable therefore the increment is also going to be a random
variable and you have n such random variables and suppose these n random variables are mutually
independent random variable for all n. So this is mixed for one n like that if you go
for all n if this property is satisfied then we can conclude the corresponding stochastic
process having the property of independent increments.
So the independent increment that does not imply some other properties but here what
we are saying is the increment satisfies the mutually independent property. That means
if you find out the CDF of the joint CDF of this random variable that is same as the product
of the individual CDF that is property satisfied by all the n then you can conclude that stochastic
process has the independent increment. The next property or the next definition is
Ergodic property. What is the meaning of Ergodic property? It says the time average of a function
along a realization or sample exists almost everywhere and is related to the space average.
What it means? Whenever the system or the stochastic process is Ergodic the time average
is the same for all almost initial points that is the process evolved for a longer time
forgets its initial state. So statistical sampling can be performed at one instant across
a group of identical processes or sampled over time on a single process with the no
change in the measured result. We will discuss the Ergodic property for the Markov process
in detail later but this Ergodic property is going to be very important when you study
the Markov property or when you study the stationarity property. Therefore these Ergodic
properties always goes along with the stationarity property or goes along with the Markov property
therefore the stochastic property is going to behave in a different way and that we are
going to discuss later. The most important stationary process that is a strict sense
stationary process. First let me start with the strict sense stationary process of order
n. Then I will define the strict sense stationary process for all order n or there is a strict
sense stationary process itself. If for arbitrary t1, t2 and so on tn the joint distribution
of the random vector that is X of t1, X of t2 and so on X of tn and the another random
vector that is X of t1 plus h comma X of t2 plus h and so on X of tn plus h are the same
for all h which is greater than 0 then we say the stochastic process
is a strict sense stationary of order n because here we restricted with the n random variable.
So we take a n random variable taken at the points t1, t2, and tn and to find out the
joint distribution of X of t1, X of t2 and X of tn so you can find out what is a joint
distribution of this n random variable. Also you find the joint distribution of n random
variable shifted by h that means earlier the random variable X of t1 now you have a random
variable X of t1 plus h with the same shift h you do it with the t2. Therefore the random
variable X of t2 plus h. Similarly the nth random variable is X of tn earlier. Now you
have a random variable X of tn plus h. So you have another random vector with n random
variables and find out the joint distribution of that. If the joint distribution of this
first n random variable as well as the joint distribution of the shifted by h that random
variable if both the distributions are same that means they are identically distributed.
The joint distributions are going to be identical then you can conclude this stochastic process
is a strict sense stochastic process of order n because you use the n random variable.
If this is going to be satisfied the above property is going to be satisfied for all
n then you can conclude the stochastic process is going to be a strict sense stationary process
for any integer n. This is going to be a strict sense stationary
process for any integer n. So we start to cross checking the joint distribution of n
random variable so if it is satisfying by only with the maximum sum integer then it
is going to be a strict sense stationary process of order that n. If it is going to be satisfied
for all n then for any integer n then it is going to be call it as just strict sense stationary
process.