On the Order and the Type of Entire Characteristic Functions

Abstract

In the theory of analytic characteristic functions (c.f.'s), it is well-known that: (1) the order of an entire c.f. cannot be less than unity, unless the function is identically equal to one (P. Levy); and (2) a necessary and sufficient condition (NASC) for a distribution function (d.f.) $F(x) \neq \epsilon (x)$ to be a "finite" d.f. is that its c.f. be an entire function of order one and of exponential type (G. Polya). (Throughout this paper, $F(x)$ will invariably denote a d.f., and $f(t)$ the corresponding c.f.) These two results lead us naturally to the investigation of conditions under which a d.f. will have an entire c.f. (i) of order one and of maximal type, or (ii) of given finite order greater than one, or (iii) of infinite order. Sufficient conditions for (ii) were obtained for absolutely continuous d.f.'s by D. Dugue [2], and for general d.f.'s by E. Lukacs (see [4], p. 142). The scope of the present paper extends beyond the problems (i), (ii) and (iii) above. In Section 6 below, we obtain NASC's for a d.f. to have an entire c.f. of given finite order greater than one. Section 7 deals with NASC's for a d.f. to have an entire c.f. of given finite order greater than one and given type (maximal, intermediate, or minimal). In Section 8, we consider entire c.f.'s of order one: we first obtain NASC's for a d.f. to have an entire c.f. of order one and maximal type; next, we obtain a relation between the extremities of a "finite" d.f. $\neq \epsilon (x)$, and the type of the corresponding entire c.f. (which is of order one); and finally, we obtain as a corollary the fact that there cannot exist an entire c.f. of order one and of minimal type. These constitute the main results of this paper. In addition to the above, we obtain in Section 3 a theorem on the moments of a d.f., and in Section 4 a theorem on the interval of existence of the integral defining the moment-generating function of a d.f., these theorems being closely related in form and in content to the main results mentioned above. Allied results on analytic c.f.'s which are not entire, and on entire c.f.'s of infinite order are given in Section 9.