0:20
Well, around the variable Xi,
has an infinitely divisible distribution
if for any natural N is larger or equals at two,
Xi can be presented as a sum of n,
identically distributed independent random variables.
So as exists, Y_1, Y_2, and so on.
Y_n a sequence of independent and identically distributed variables,
such as Xi equal distribution to the sum.
Well, in terms of the characteristic functions,
this is exactly the same as to say there is
a characteristic function of the random variable of Xi is
equal to the characteristic function of
the random variable Y_1 is at U and as a power n. Okay.
It's completely clear that this statement is exactly the same as this one.
In other words, besides,
infinitely divisible if and only if the N-th root of F, XIFU,
is also a characteristic function
for any N. Well,
this class is rather interesting and for us,
most important property of this clause is that this has
some very close relation to the clause of Levy processes.
Namely this relation can be reviewed by the following proposition.
3:27
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.
9:18
Well and it turns out that
Cauchy distribution is also an infinitely divisible distribution.
It can be easily seen from this property of the characteristic function because phi of u.
In this case is equal to exponent is the power x zero
iu minus gamma multiplied by the absolute value of u.
And if you calculate the nth root of this function,
we get that it is equal to exponent in the power x0 divided by n,
multiplied by iu minus gamma divided by n multiplied by square root of u.
So we have as the nth root of this function has also Cauchy
distribution with bar matrix x0 divided by n and
gamma divided by n. So what we
have here that the Cauchy distribution is also infinitely divisible distribution.
Let me provide some further examples.
Well, let me consider the so-called gamma distribution with bar matrix alpha and beta,
alpha and beta and two positive numbers.
This distribution, the distribution with the following density of
p of x is equal to beta is
a power alpha divided by gamma function at the point alpha multiplied by x
to the power of minus one and multiplied by
exponent as the power minus beta x for positive x.
The distribution is supported only on positive half line and this is
the difference between this distribution and
two examples which are considered before namely
Normal distribution and Cauchy distribution.
Well, here we'll have two parameters often beta and this parameters have clear meaning.
Basically the skewness and the kurtosis of
this distribution are completely determined by the parameter of
more precisely skewness is
equal to two divided by the square root of alpha and as for kurtosis,
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it's equal to six divided by alpha.
And the parameter of beta has a meaning of a scale parameter.
The sense that if x has a gamma distribution with
parameters alpha and beta and lambda is some positive number then
x divided by lambda has a gamma distribution with
parameters alpha and beta multiplied by lambda.
And this situation naturally arises when you change the scale.
So for instance you measured x in metres and now you would like to
measure it in other, whereas in centimeters.
Okay. And as for the characteristic function of the gamma distribution,
it is equal to one minus iu divided by beta in the power minus alpha.
Additional calculate the nth root of this characteristic function.
You get exactly the same form of
the characteristic function but with alpha and an n instead of alpha and therefore,
the nth root of the characteristic function has also
a gamma distribution with parameters alpha divided by n and beta.
So the gamma distribution is another example of an infinitely divisible distribution.
And these examples are interesting because
this distribution is not of the whole real line but only of the positive half line.
Let me now summarize all examples of
infinitely divisible distributions which are widely used.
I will provide at list of infinitely divisible distributions.
First of all examples which you can see that already.
This is a normal distribution and Cauchy distribution.
Then we have gamma distribution
and exponential distribution.
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Exponential distribution is actually a particular case of a gamma distribution.
So we have the so-called negative binomial distribution
and geometric distribution.
Both of these distributions are related to
the Bernoulli scheme and I will also combine them.
Also the Poisson process is a another process and
Poisson distribution is an infinitely divisible distribution and
also a compound Poisson Process is
another example of a Levy Process and there is a notion of
compound Poisson distribution which basically means that
the characteristic function of this there in
the variable with the distribution is of the form exponent in
the power lambda multiplied by another characteristic function minus one.
Okay. We're all from this type
of distributions is mixed with this so called stable distributions.
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The definition of a stable distribution is a fooling one.
We say that Psi has a stable distribution,
if for any n sum of
n independent identically distributed render variables
with the the same distribution as Psi
possess the following property: It is equal to An multiplied by Psi plus Bn where
An and Bn are two deterministic sequences
depending on n. The main difference between these type
of distributions and this one is that here y1 and yn may
have another distribution than Psi and here they should have the same.
There is no doubt any stable distribution is often infinitely divisible distribution
but the converse is not true.
And basically from this list,
there are only two examples of distributions which are stable,
this are Normal and Cauchy.
All other distributions are not stable.
I believe that this list of examples is rather long and I
would like to stop here in the next subsection.
I will provide a couple of examples which are not infinitely divisible.
And to show that they are not from this class,
I will formulate a couple of properties which hold for any for
any infinitely divisible distribution and I will show that
this counter examples do not satisfy them.
Vladimir PanovAssistant Professor
