Journal of the London Mathematical Society, Dec 1, 1983
We investigate global properties of the solution of the delay differential equation x/'(x) = (A-1... more We investigate global properties of the solution of the delay differential equation x/'(x) = (A-1)/(*)-£ kj(x-p t) i = l under the conditions that f(x) = 0 for x < 0, /(p,) = c 0 (> 0), 0 < p t < p 2 < ... < p n < oo, k > 0 and A, > 0 for i = 1, 2,..., n, and prove that the solution f(x) is positive on (0, oo) and integrable over (0, oo) if n and only if k = £ A f. In probability theory (cf. [2]) a distribution function F(x) is said to be unimodal with mode a if F(x) is convex for x < a and concave for x > a; for short F(x) is said to be unimodal. S. J. Wolf [6] proved that the probability density of a special type of distribution function of class L satisfies the delay differential equation (DDE)-£ XJ(x-Pi), l) = c 0 (> 0) , 0 < Pi < Pi < •" < Pn < °° , > 0 , A,-> 0 (i = 1,2 n), and that its distribution function F(x) = f(t)dt is unimodal. By using the facts J-oo that /(x) is continuous and non-negative on (0, oo) and that /(x)-> 0 as x-> oo and /(x) = 0 (x < 0) because f(x) is the probability density of the distribution function of class L, S. J. Wolf could prove the unimodality. In this paper we investigate global properties of the solution /(x) of the equation (•) x/'(x) = (A-l)/(x)-t lif(x-Pi) i = l
In this paper we consl'der a probability distribution whose density is the normed product of Cauc... more In this paper we consl'der a probability distribution whose density is the normed product of Cauchy densities such as a following form,
This paper is to prove the infinite divisibility of a t distribution with the odd degrees of free... more This paper is to prove the infinite divisibility of a t distribution with the odd degrees of freedom 2n + 1 without using of Bessel functions. The Laplace-Stieltjes transform of an infinitely divisible probability distribution is concentrated on the interval [0, 1). A simple expression of the Laplace-Stieltjes transform is made by using of hypergeometric function. The L´evy measure of the Student t distribution obtained from the L´evy mesure of the distribution with the density functon.
Journal of the London Mathematical Society, Dec 1, 1983
We investigate global properties of the solution of the delay differential equation x/'(x) = (A-1... more We investigate global properties of the solution of the delay differential equation x/'(x) = (A-1)/(*)-£ kj(x-p t) i = l under the conditions that f(x) = 0 for x < 0, /(p,) = c 0 (> 0), 0 < p t < p 2 < ... < p n < oo, k > 0 and A, > 0 for i = 1, 2,..., n, and prove that the solution f(x) is positive on (0, oo) and integrable over (0, oo) if n and only if k = £ A f. In probability theory (cf. [2]) a distribution function F(x) is said to be unimodal with mode a if F(x) is convex for x < a and concave for x > a; for short F(x) is said to be unimodal. S. J. Wolf [6] proved that the probability density of a special type of distribution function of class L satisfies the delay differential equation (DDE)-£ XJ(x-Pi), l) = c 0 (> 0) , 0 < Pi < Pi < •" < Pn < °° , > 0 , A,-> 0 (i = 1,2 n), and that its distribution function F(x) = f(t)dt is unimodal. By using the facts J-oo that /(x) is continuous and non-negative on (0, oo) and that /(x)-> 0 as x-> oo and /(x) = 0 (x < 0) because f(x) is the probability density of the distribution function of class L, S. J. Wolf could prove the unimodality. In this paper we investigate global properties of the solution /(x) of the equation (•) x/'(x) = (A-l)/(x)-t lif(x-Pi) i = l
In this paper we consl'der a probability distribution whose density is the normed product of Cauc... more In this paper we consl'der a probability distribution whose density is the normed product of Cauchy densities such as a following form,
This paper is to prove the infinite divisibility of a t distribution with the odd degrees of free... more This paper is to prove the infinite divisibility of a t distribution with the odd degrees of freedom 2n + 1 without using of Bessel functions. The Laplace-Stieltjes transform of an infinitely divisible probability distribution is concentrated on the interval [0, 1). A simple expression of the Laplace-Stieltjes transform is made by using of hypergeometric function. The L´evy measure of the Student t distribution obtained from the L´evy mesure of the distribution with the density functon.
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