在零点的隣區內彼此相等的特徵函数

<正> §1.引言 大家知道,兩個不相恆等的特徵函數(以下简称特函)可以在零點的隣區內相等。為固定用語起見,在本文中我們說特函f(t)属於集合(U),如果存在一個特函,它与f(t)在零的隣區內相等,但並不恆等於f(t);如果f(t)不屬於(U),就說它屬於(U)。

ON CHARACTERISTIC FUNCTIONS WHICH COINCIDE IN THE NEIGHBORHOOD OF ZERO

We say in this paper that a characteristic function (c.f.) belongs to the class (U), if it can equal to another c.f. in the neighborhood of zero without equalling it identically. A c.f. is said to belong to (U) if it does not belong to (U).Summing up the known examples of c.f.'s belonging to (U), we state the following result: If g(t) is a real-valued even function, with g(o) = 1, which is convex, non-negative and non-increasing on the positive t-axis, then g(t) is α c.f. and (unless g(t)=1) belongs to (U).It is found that the c.f.'s pertaining to a sub-class of stable laws; exp { - (1- ic sgn t) |t|~α} (A) belong to (U) The precise result is: For every 0<α<1, there exists a positive constant b, depending on α only, such that for all |c| ≤b the c.f. (A) belongs to (U).In an attempt to discover c.f.'s which belong to (U) and which are differentiable an unlimited number of times we find the following theorem helpful:Let q(x) be summable and Hermitian on the whole x-axis, and not equivalent to zero. Let the Fourier transform of q(x) vanish on an interval. Then the probability density function (p.d.f.) |q(x)| has its c.f. belonging to (U). We notice in passing that the p.d.f's |q(x)|=(1+x~2)~(-λ)(λ>1;q(x)=(1-ix)~(-λ)) and |q(x)| = |x|~(-n) | sin x|~n(n=2,3,4,…,q(x) = x~(-n) sin x)~n)are of this type.This theorem has the following consequence:Let θ(t) be a real and measurable function on (0,∞), with and let Let e~(-f(x)) be summable over (-∞,∞). Then the p.d.f, e~(-f(x)) has its c.f. belonging to (U).By taking θ(t) = α (1-α) t~(α-2) (0 < α < 1) we obtain the p.d.f. exp(-|x|~α). It can be shown that a proper choise of θ(t) will give the p.d.f.p(x)=c~(-|x|), 0≤|x|≤A;p(x)=e~(-|x|/ψ(|x|)),|x|>A, where ψ(x) is any member of the hierarchy (ln x)~λ, (ln x)(ln.ln x)~λ,…, (λ>1), and where A is determined by ψ(A)=1. It is an open question to discover a p.d.f. whose tails are even smaller than those of p(x) and whose c.f. belongs to (U).The rest of the paper is devoted to two theorems about the convergence of a sequence of c.f's in the neighborhood of zero. They are stated as follows:Theorem 6. Let {F_n(x)}be a sequence of distribution functions, with the corresponding sequence of c.f.'s {f_n(t)}. If f_n(t) tends to a limit h(t) for |t|<δ, and if the function h(t) is continuous at t=0, then the sequence { F_n(x) }is compact (i.e. every convergent sub-sequence of it converges to a distribution function).Theorem 7. Concerning any c.f. f(t) the following two statements are equivalent:1°f(t) belongs to (U);2°any sequence of c.f.'s which converges to f(t) in the neighborhood of zero must converge to it for all t.With the help of these two last theorems we are able to show that a recent theorem of Zygmund (Second Berkeley Symposium (1951, pp. 369-372) is a consequence of a. result of Marcinkiewicz (Fund. Math., 31 (1938), pp. 86-102) to the effect that a sufficient condition for a c.f. to belong to (U) is that the corresponding distribution function F(x) should satisfy the condition where r is certain positive constant.