Almost sure convergence of measure-valued branching processes: a critical exponent

1994 ◽  
Vol 124 (4) ◽  
pp. 811-823
Author(s):  
Alison M. Etheridge
Keyword(s):  
Markov Process ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Upper And Lower Bounds ◽  
Spatial Process ◽  
Sufficient Condition ◽  
Time Process ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Occupation Time Process

A large class of measure-valued critical branching processes can be classified in terms of a parameter ρ which arises as a measure of the recurrence of the underlying spatial Markov process. By establishing upper and lower bounds for the total weighted occupation time process, it is shown that if a measure-valued process is started from an invariant measure of its underlying spatial process, then a necessary and sufficient condition for (a.s.) local extinction is that ρ > 0.

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.

Almost sure convergence in Markov branching processes with infinite mean

1977 ◽  
Vol 14 (04) ◽  
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.

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.

Capacity Estimates for Planar Cantor-Like Sets

1974 ◽  
Vol 26 (5) ◽  
pp. 1169-1172
Author(s):  
Carl David Minda
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Elementary Approach ◽  
Principal Tool ◽  
Linear Problems ◽  
Necessary And Sufficient ◽  
Positive Capacity ◽  
Capacity Estimates

Upper and lower bounds for the capacity of planar Cantor-like sets are presented. Chebichev polynomials are the principal tool employed in the derivation of these estimates. A necessary and sufficient condition for certain planar Cantor-like sets to have positive capacity is obtained. Related one-sided capacitary estimates for more general Cantor-like sets can be found in [3, pp. 106-109]. Techniques analogous to those used in this paper yield similar results for linear Cantor-like sets which are well-known [2, pp. 150-161]. The use of Chebichev polynomials to obtain these results provides an alternate, possibly more elementary, approach to these linear problems.

scholarly journals A Glivenko-Cantelli Bootstrap Theorem for the Foster-Greer-Thorbecke Poverty Index

2021 ◽  
Vol 54 (2) ◽  
pp. 161-179
Author(s):  
Pedro A. Harmath ◽  
Josefa Ramoni-Perazzi ◽  
Abelardo Monsalve-Cobis
Keyword(s):  
Empirical Process ◽  
Almost Sure Convergence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Poverty Index ◽  
Necessary And Sufficient

Abstract.We assume the Foster-Greer-Thorbecke (FGT) poverty index as an empirical process indexed by a particular Glivenko-Cantelli class or collection of functions and define this poverty index as a functional empirical process of the bootstrap type, to show that the outer almost sure convergence of the FGT empirical process is a necessary and sufficient condition for the outer almost sure convergence of the FGT bootstrap empirical process; that is: both processes are asymptotically equivalent respect to this type of convergence.

Three-terminal communication channels

1971 ◽  
Vol 3 (1) ◽  
pp. 120-154 ◽  
Author(s):  
Edward C. Van Der Meulen
Keyword(s):  
Mutual Information ◽  
Lower Bounds ◽  
Communication Channel ◽  
Communication Channels ◽  
Upper And Lower Bounds ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Symmetric Structure ◽  
Necessary And Sufficient ◽  
Positive Capacity

The problem of transmitting information in a specified direction over a communication channel with three terminals is considered. Examples are given of the various ways of sending information. Basic inequalities for average mutual information rates are obtained. A coding theorem and weak converse are proved and a necessary and sufficient condition for a positive capacity is derived. Upper and lower bounds on the capacity are obtained, which coincide for channels with symmetric structure.

Three-terminal communication channels

1971 ◽  
Vol 3 (01) ◽  
pp. 120-154 ◽  
Author(s):  
Edward C. Van Der Meulen
Keyword(s):  
Mutual Information ◽  
Lower Bounds ◽  
Communication Channel ◽  
Communication Channels ◽  
Upper And Lower Bounds ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Symmetric Structure ◽  
Necessary And Sufficient ◽  
Positive Capacity

The problem of transmitting information in a specified direction over a communication channel with three terminals is considered. Examples are given of the various ways of sending information. Basic inequalities for average mutual information rates are obtained. A coding theorem and weak converse are proved and a necessary and sufficient condition for a positive capacity is derived. Upper and lower bounds on the capacity are obtained, which coincide for channels with symmetric structure.

On branching processes allowing immigration

1972 ◽  
Vol 9 (01) ◽  
pp. 24-31 ◽  
Author(s):  
Y. S. Yang
Keyword(s):  
Invariant Measure ◽  
Discrete Time ◽  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Recurrence ◽  
Discrete Time Case ◽  
Necessary And Sufficient

Continuous time one-type branching processes allowing immigration are considered. The invariant measure, which is shown to be unique, is exhibited. From this, a condition for positive recurrence similar to that of Heathcote's in the discrete time case is obtained. For the critical discrete time case, Seneta's sufficient condition for positive recurrence is improved to give a necessary and sufficient condition.

On branching processes allowing immigration

1972 ◽  
Vol 9 (1) ◽  
pp. 24-31 ◽  
Author(s):  
Y. S. Yang
Keyword(s):  
Invariant Measure ◽  
Discrete Time ◽  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Recurrence ◽  
Discrete Time Case ◽  
Necessary And Sufficient

Continuous time one-type branching processes allowing immigration are considered. The invariant measure, which is shown to be unique, is exhibited. From this, a condition for positive recurrence similar to that of Heathcote's in the discrete time case is obtained. For the critical discrete time case, Seneta's sufficient condition for positive recurrence is improved to give a necessary and sufficient condition.

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