Infinitely divisible distributions. Normal law. Multidimensional distributions

L. D. Meshalkin
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L. D. Meshalkin: Moscow State University

Chapter 7 in Collection of problems in probability theory, 1973, pp 96-106 from Springer

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

Keywords: Characteristic Function; Preceding Problem; Divisible Distribution; Arbitrary Collection; Positive Definite Quadratic Form (search for similar items in EconPapers)
Date: 1973
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DOI: 10.1007/978-94-010-2358-0_7

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