Dispersive ordering by dilation

In this paper a necessary and sufficient condition for the dispersive ordering in dilation sense is given by a convex function which is called the dispersive function and characterizes the distribution function. Some interesting properties of the ordering follow from this result.

Download Full-text

On the convergence of the supercritical branching processes with immigration

Journal of Applied Probability ◽

10.2307/3213010 ◽

1977 ◽

Vol 14 (2) ◽

pp. 387-390 ◽

Cited By ~ 3

Author(s):

Harry Cohn

Keyword(s):

Distribution Function ◽

Limit Distribution ◽

Branching Process ◽

Branching Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Limit Distribution Function ◽

Branching Process With Immigration ◽

Necessary And Sufficient ◽

Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn} exists such that {Xn/cn} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn} such that {Xn/cn} converges in law to a proper limit distribution function F, with F(0 +) < 1.

Download Full-text

Subexponential distribution functions

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700021340 ◽

1980 ◽

Vol 29 (3) ◽

pp. 337-347 ◽

Cited By ~ 90

Author(s):

E. J. G. Pitman

Keyword(s):

Distribution Function ◽

Comparison Theorem ◽

Distribution Functions ◽

Sufficient Condition ◽

Important Class ◽

Necessary And Sufficient Condition ◽

Closure Properties ◽

Subexponential Class ◽

Necessary And Sufficient ◽

60 J 80

AbstractA distribution function (F on [0,∞) belongs to the subexponential class if and only if 1−F(2) (x) ~ 2(1−F(x)), as x→ ∞. For an important class of distribution functions, a simple, necessary and sufficient condition for membership of is given. A comparison theorem for membership of and also some closure properties of are obtained.1980 Mathematics subject classification (Amer. Math. Soe.): primary 60 E 05; secondary 60 J 80.

Download Full-text

Bi-Positive Sequences the Bilateral Moment Problem

Canadian Mathematical Bulletin ◽

10.4153/cmb-1986-072-8 ◽

1986 ◽

Vol 29 (4) ◽

pp. 456-462 ◽

Cited By ~ 2

Author(s):

Jaime Vinuesa ◽

Rafael Guadalupe

Keyword(s):

Distribution Function ◽

Moment Problem ◽

General Setting ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

AbstractWe pose a “moment problem” in a more general setting than the classical one. Then we find a necessary and sufficient condition for a sequence to have a solution of the “problem“where σ is a “distribution function”.

Download Full-text

Bounds on Texture Coefficients

Journal of Applied Mechanics ◽

10.1115/1.1533808 ◽

2003 ◽

Vol 70 (2) ◽

pp. 200-203 ◽

Cited By ~ 1

Author(s):

J. C. Nadeau ◽

M. Ferrari

Keyword(s):

Distribution Function ◽

Orientation Distribution Function ◽

Orientation Distribution ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Velocity Measurements ◽

Texture Coefficients ◽

Generalized Spherical Harmonics ◽

Necessary And Sufficient ◽

Microstructural Optimization

The orientation distribution function (ODF) is expanded in terms of generalized spherical harmonics and bounds on the resulting texture coefficients are derived. A necessary and sufficient condition for satisfaction of the normalization property of the ODF is also provided. These results are of significance in, for example, microstructural optimization of materials and predicting texture coefficients based on wave velocity measurements.

Download Full-text

On the convergence of the supercritical branching processes with immigration

Journal of Applied Probability ◽

10.1017/s0021900200105078 ◽

1977 ◽

Vol 14 (02) ◽

pp. 387-390 ◽

Cited By ~ 1

Author(s):

Harry Cohn

Keyword(s):

Distribution Function ◽

Limit Distribution ◽

Branching Process ◽

Branching Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Limit Distribution Function ◽

Branching Process With Immigration ◽

Necessary And Sufficient ◽

Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn } exists such that {Xn/cn } converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn } such that {Xn /c n} converges in law to a proper limit distribution function F, with F(0 +) < 1.

Download Full-text

Almost sure convergence in Markov branching processes with infinite mean

Journal of Applied Probability ◽

10.2307/3213344 ◽

1977 ◽

Vol 14 (4) ◽

pp. 702-716 ◽

Cited By ~ 14

Author(s):

D. R. Grey

Keyword(s):

Functional Equation ◽

Distribution Function ◽

Continuous Time ◽

Branching Process ◽

Branching Processes ◽

Almost Sure Convergence ◽

Random Variable ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

Download Full-text

Almost sure convergence in Markov branching processes with infinite mean

Journal of Applied Probability ◽

10.1017/s0021900200105248 ◽

1977 ◽

Vol 14 (04) ◽

pp. 702-716 ◽

Cited By ~ 7

Author(s):

D. R. Grey

Keyword(s):

Functional Equation ◽

Distribution Function ◽

Continuous Time ◽

Branching Process ◽

Branching Processes ◽

Almost Sure Convergence ◽

Random Variable ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

If {Zn } is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn } and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt } with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

Download Full-text

THEORY OF THE GOOS-HÄNCHEN DISPLACEMENT IN TOTAL INTERNAL REFLECTION

International Journal of Modern Physics B ◽

10.1142/s0217979207037326 ◽

2007 ◽

Vol 21 (16) ◽

pp. 2777-2791 ◽

Cited By ~ 8

Author(s):

JIE-LONG SHI ◽

CHUN-FANG LI ◽

QI WANG

Keyword(s):

Distribution Function ◽

Total Internal Reflection ◽

Incident Beam ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Dielectric Interface ◽

Parallel Component ◽

Spectral Distribution Function ◽

Position Prediction ◽

Necessary And Sufficient

Since the Goos-Hänchen (GH) effect is the displacement of the totally reflected beam at a dielectric interface from the position prediction by geometrical reflection, the concept of GH displacement is applicable only when the reflected beam retains the shape of the geometrically reflected or incident beam. The necessary and sufficient condition has been advanced for the totally reflected beam to retain the shape of the incident beam. Numerical simulations have been performed to confirm this condition. It has been shown that the GH displacement results from the mechanism of beam reshaping due to the linear dependence of the reflection phase shift upon the parallel component of the wave vector, in the interval in which the angular spectral distribution function of the incident beam is appreciable.

Download Full-text

The Identification of Convex Function on Riemannian Manifold

Mathematical Problems in Engineering ◽

10.1155/2014/273514 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Li Zou ◽

Xin Wen ◽

Hamid Reza Karimi ◽

Yan Shi

Keyword(s):

Riemannian Manifold ◽

Convex Function ◽

Global Minimum ◽

Directional Derivative ◽

Inequality Constraints ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Generalized Gradient ◽

Necessary And Sufficient ◽

Global Minimum Point

The necessary and sufficient condition of convex function is significant in nonlinear convex programming. This paper presents the identification of convex function on Riemannian manifold by use of Penot generalized directional derivative and the Clarke generalized gradient. This paper also presents a method for judging whether a point is the global minimum point in the inequality constraints. Our objective here is to extend the content and proof the necessary and sufficient condition of convex function to Riemannian manifolds.

Download Full-text