L_t multiplied by K divided by N minus L_t multiply by K minus one divided by
N. And you see it that here will have the increments of
Levy process L_t from
some intervals which are not intersected and therefore this increments are independent.
And more over, since their distribution
depends only on the difference between the arguments.
Well, the different has exactly the same distribution as L multiplied,
L is the moment T divided by N. And so therefore will
have a sequence of the dependent identically distributed trend or variables
and LT is equal to the sum of all other elements from this segments.Therefore exactly
this property is fulfills that this L_t at
any time moment has an infinitely divisible distribution.
The second part is much more complicated,
and don't want to prove it.
But this part is very important because tutor's statement,
class of Libya prothesis and place of
infinitely divisible distributions are essential as I said.
So, to determine differently divisible distribution
is the same as to determine a Levy process.
Okay.
Now, let's provide some examples of the distributions which are in this class.
So first example, is a normal distribution whose perimeter Mu and sigma squared.
Well, around a variable Xi having it's distribution,
can be presented as the sum of an independent identical distributive trend of variables,
where each independent variable YK
has a normal distribution as per matters Mu divided by N,
and sigma squared divided by N. This is a simple corollary from the fact that
some of independent normally distributed random variables has also normal distribution,
and the mean size equal to the sum of
the means and variance is equal to the sum of variances.
So this is actually nothing more than application or for
a simple ideas of probability theory.
On the other side,
we can employ this alternative definition to show exactly the same effect,
because the artistic function of Xi,
is equal to exponent is power
I Mu minus one a half sigma squared, U squared.
And if now we calculate the Nth root of FiXi of U,
we would get it that is equal to exponent is the power I Mu divided by N U,
minus one half sigma squared divided by N U squared.
So will have exactly the same answer.
So it is also normal distribution which mean Mu divided by
N and variance sigma squared divided by L. We get
the same answer but using another approach.
So finally, we conclude that
the normal distribution is infinitely divisible distribution,
and actually it corresponds to the Brownian motion with drift that
this is a process which is equal to Mu T plus sigma W.T,
where W is a Brownian motion.
This is Levy process which has at the time moment one exactly this distribution.
Okay, this is our first example of Levy process we have
shown that the sequence why one way to end so
white hand can be found as from the definition and from this property.
Well let's improvise some further examples.
Well one more example is given by a cost sheet distribution.
Cauchy distribution.
This is the distribution was a full length density by a pair
of X is equal to one divided by pi Gamma,
multiplied by one plus X minus X zero squared divided by Gamma squared.
The distribution is known as the simplest example
of a distribution result, Mathematical expectation.
And this bar matrix Gamma and then zero have some clear mean.
So x zero is a location parameter.
If i draw a plot of this density function, so exactly.
Well, to the maximum was his functions I think he's given by x zero.
And in another parameter of Gamma is a scale parameter,
when Gamma is larger than the maximum, is also higher.
