Infinitely divisible distributions. Normal law. Multidimensional distributions

Abstract

The only new concept in this chapter is that of the infinitely divisible (i.d.) distribution law; in this connection, see Chapter 9 of the textbook by B. V. GNEDENKO. The distribution law F(x) is called i.d. if its characteristic function, for an arbitrary integer n ≥ l, can be written in the form

$$f(t) = [f_n (t)]^n ,$$

where f _n (t) is also a characteristic function. In Problems 375, 381–387, it is assumed that the general form of the logarithm of the characteristic function of the i.d. law

$$\log f(t) = iyt + \int\limits_{ - \infty }^\infty {\left( {e^{itu} - 1 - \frac{{itu}}{{1 + u^2 }}} \right)} \frac{{1 + u^2 }}{{u^2 }}\text{dG}(u),$$

((1))

is knowm, where G(u) is a nondecreasing function of bounded variation, and the function under the integral sign is defined by the equality

$$\left[ {\left\{ {\left. {e^{itu} - 1 - \frac{{itu}}{{1 + u^2 }}} \right\}\frac{{1 + u^2 }}{{u^2 }}} \right.} \right]_{u = 0} = - \frac{{t^2 }}{2} $$

for u = 0. It is also assumed known that the representation of log f(t) by formula (1) is unique.