Now, let me consider some more general case when we have a compound Poisson process and the distribution of Xi1, Xi2 and so on can be arbitrary. In this case, we should employ the so-called characteristic functions. Let me define this. So, for random variable Xi, its characteristic function is defined as the mathematical expectation of the exponent in the power complex unit uXi. So, characteristic function is a function which maps real r into complex values. Now, if you have two independent different variables, Xi1 and Xi2, then the characteristic function of the sum is equal to the product of two characteristic functions, Xi1 and Xi2. This is a very important property and you know that for moment-generating functions, probability generating functions, this property also holds. Nevertheless, we need exactly these kinds of transformations of random variables Xi, because it is currently defined for any random variable Xi. You know that the probability generating function is correlative defined only for integer valued non-negative random variables, as well as Laplace transform is correlative defined only for positive or non-negative random variables. And here, you can introduce this object for any Xi, and it is actually finite for any random real u. Let me employ this kind of transformations to our case, to the case of compound Poisson processes. Basically, the following theorem holds. The theorem states that Xt, has independent and stationary increments and moreover, the characteristic function of the increment Xt minus Xs at the point u is equal to the exponent in the power lambda t minus s, characteristic function of one [inaudible] Xi1 minus u, minus one. This formula holds for any t larger than s larger or equal to zero. This is a very important property of the compound Poisson processes and you can actually derive many correlates only from this formula, and therefore I would like to show why it is true. Let me provide at least a sketch as a proof. Let me consider the left-hand side of this equality. So, it is equal to the mathematical expectation of the exponent in the power iu (Xt minus Xs). This exponent, this expectation can be considered by a kind of total probability law and therefore, it is equal to the sum k from zero to infinity, mathematical expectation of exponent in the power iu (Xt minus Xs) given that Nt minus Ns is equal to k multiplied by the probability that Nt minus Ns equal to k. Now, what we have here inside this expectation, what is Xt minus Xs if you know that Nt minus Ns is equal to k? If you will think about this, you will immediately realize it is actually the sum of Xi1 and so on, Xik. At least in this distribution, this is equal to this sum. And if you replace here, Xt minus Xs by this sum, you will get that this function depends on Xi and you take a condition which depends on N. Since Xi and N are independent, this conditional expectation is equal to unconditional. You can simply cross this condition. And this force a probability that Nt minus Ns is equal to k. You know that Nt minus Ns has a Poisson distribution with parameter lambda t minus s, and therefore you know the exact expression for this probability. Now, you should employ the property of the characteristic function. In fact, you have a sum of k independent identical distributed random variables, and therefore the characteristic function of the sum is equal to the product of k characteristic function, which are equal to each other. Finally, you get that this sum is equal to the sum by k from zero to infinity, characteristic function of Xi1 u in the power k multiplied by this probability, it is equal to the exponent in the power minus lambda t minus s multiplied by lambda t minus s in the power k divided by k factorial. Finally, if you will think about this sum and deploy that exponent is equal to the sum x in the power N divided by N factorial sum from N from zero to infinity, you will get that this expression exactly equals this expression which we showed proof. So, this observation completes the proof and now, I would like to show how we can apply this theorem for showing various properties of the compound Poisson process.