Let me introduce the class of Infinitely Divisible Distributions. 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. So, this proposition has two parts. First of all, any levy process, L_t, at any time moment, T has an infinitely divisible distribution. And second part, for any infinitely divisible distribution, there exists a levy process, L_t such that this process at some time moment has this distribution name for instance. One has this distribution. So first part of this proposition can be easily drawn from the definition of Levy process. In fact the process L_t for any end can be represented as follows. We can say that L_t is equal to the sum K from one to N, 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. 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, 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. 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. 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.
