Subexponential distribution functions

1980 ◽  
Vol 29 (3) ◽  
pp. 337-347 ◽  
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.

A Classification and Closure Properties of Languages for Describing Concurrent System Behaviours

1981 ◽  
Vol 4 (3) ◽  
pp. 531-549 ◽  
Author(s):  
Miklós Szijártó
Keyword(s):  
Formal Languages ◽  
Parallel Program ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Concurrent System ◽  
Closure Properties ◽  
Sequential Program ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Independence Relations

The correspondence between sequential program schemes and formal languages is well known (Blikle and Mazurkiewicz (1972), Engelfriet (1974)). The situation is more complicated in the case of parallel program schemes, and trace languages (Mazurkiewicz (1977)) have been introduced to describe them. We introduce the concept of the closure of a language on a so called independence relation on the alphabet of the language, and formulate several theorems about them and the trace languages. We investigate the closedness properties of Chomsky classes under closure on independence relations, and as a special case we derive a new necessary and sufficient condition for the regularity of the commutative closure of a language.

On the convergence of the supercritical branching processes with immigration

1977 ◽  
Vol 14 (2) ◽  
pp. 387-390 ◽  
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.

Some generalized variability orderings among life distributions with reliability applications

1991 ◽  
Vol 28 (02) ◽  
pp. 374-383 ◽  
Author(s):  
M. C. Bhattacharjee
Keyword(s):  
Random Variables ◽  
Parallel Systems ◽  
Reliability Theory ◽  
Dominance Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closure Properties ◽  
Life Distributions ◽  
The Mean ◽  
Necessary And Sufficient

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

Dispersive ordering by dilation

1990 ◽  
Vol 27 (02) ◽  
pp. 440-444 ◽  
Author(s):  
J. Muñoz-Perez ◽  
A. Sanchez-Gomez
Keyword(s):  
Distribution Function ◽  
Convex Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dispersive Ordering ◽  
Necessary And Sufficient

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.

Bi-Positive Sequences the Bilateral Moment Problem

1986 ◽  
Vol 29 (4) ◽  
pp. 456-462 ◽  
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”.

Bounds on Texture Coefficients

2003 ◽  
Vol 70 (2) ◽  
pp. 200-203 ◽  
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.

On the convergence of the supercritical branching processes with immigration

1977 ◽  
Vol 14 (02) ◽  
pp. 387-390 ◽  
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 &lt; m &lt; ∞ 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 +) &lt; 1.

Almost sure convergence in Markov branching processes with infinite mean

1977 ◽  
Vol 14 (4) ◽  
pp. 702-716 ◽  
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.

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