A remark on the smoothness of the distributed function.
Hajime Makabe · Medical Entomology and Zoology · 1960
\S 1. In \S 2, we shall give some relations between the smoothness of the symmetrized distribution function $(d.f.)P(x)$ of a random variable $X$ i.e. Lipschitz order of continuity of $F(x)$ , and $F(x)$ and the integrability of the characteristic function $(c.f.)f(t)=\int_{-\infty}^{\infty}e^{\ell px}dF(x)$ at infinity, and also discuss similar properties concerning the mean concentration function introduced by Prof. T. Kawata in [1). In \S 3, we shall also treat a theorem on the smoothness of $F(x)$ itself from a similar view point which complements the theorems in \S 2. \S 2. Prof. Kawata introduced the mean concentration function of a random variable $X$ as follows [1], [2]: