The last part of this lecture will be devoted to compound Poisson processes. This is a very popular model which is essentially based on what you call homogeneous Poisson processes. And according to this model, the process is defined as follows. This is the sum by k from one to some Poisson process Nt of eSk. Here random variables eS1, eS2, and so on are independent and identically distributed, not necessarily positive. The process Nt is Poisson. I mean homogeneous Poisson and I will denote its intensity parameter by lambda. And also, assumes that eS1, eS2, and so on and Nt are independent. This model eventually arises in many applications, but a lot of applications can be found in insurance. In fact, Xt can be used for modeling the aggregated claim amounts of one insurance company. According to this interpretation eS1, eS2, and so on are actually individual claim sizes. And Nt expresses the amount of claims till time t. Once more, under this interpretation, Xt is an aggregated claim amount. For analyzing of this processes, we'll need to employ some functionals of this process. So, here the main difficulty is the same as in the case of renewal processes. The point is that the distribution of Xt can be written in closed form, only in few cases which are rather trivial. In general case, it is unknown and we should use some functionals and the point is, that a kind of those functional definitely depends on the properties of the random variables eS1, eS2, and so on. More precisely, if eS1, eS2, and so on have only integer values which are non-negative. Then Xt is also an integer non-negative random variable. And in this case we can employ the so called probability generating function. So, the so-called BGF. This function is defined as follows. Actually, for a random variable eS which has only integer non-negative values. This probability generating function is defined as a mathematical expectation of u in the power of eS. Where this deterministic number, u is smaller than one. Absolutely is smaller than one. Of course, if eS1 and eS2 are independent. Then, the probability generating function of their sum using the fact a product of their corresponding generating function. In more general case, when eS1 and eS2 and so on on the negative solves it can be non-integer, but there no negative. Xt is also non-negative. And in this case one can imply the so called, moment generating function. Which is very close to the so-called Laplace transform. Let me define it as mathematical expectation of the exponent minus u in the of power eS. So, once more this definition is correct for NeS taking non-negative values and the parameter u here is positive. Sometimes, you can find in the books here plus without minus. From my point of view it's a question of taste. But here I defined it in this way just because to be sure it's object this. In fact, if you look attentively as a probability generating function, we get that this object exists for eS from this question for u from this range. The same situation is for moment generating function. Nevertheless, it seems very natural to think that eS can take any values. And from this case, we should apply the so-called characteristic function.