scholarly journals A necessary and sufficient condition for noncertain extinction of a branching process in a random environment (BPRE)

1984 ◽  
Vol 16 (2) ◽  
pp. 189-197 ◽  
Author(s):  
John Coffey ◽  
David Tanny
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

A note on the subcritical generalized age-dependent branching process

1976 ◽  
Vol 13 (4) ◽  
pp. 798-803 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

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.

A heterogeneous blocking system in a random environment

1996 ◽  
Vol 33 (01) ◽  
pp. 211-216 ◽  
Author(s):  
G. Falin
Keyword(s):  
Random Environment ◽  
Joint Distribution ◽  
Service System ◽  
External Environment ◽  
The State ◽  
Product Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We obtain a necessary and sufficient condition for the interaction between a service system and an external environment under which the stationary joint distribution of the set of busy channels and the state of the external environment is given by a product-form formula.

A note on the age-dependent branching process with immigration

1975 ◽  
Vol 12 (01) ◽  
pp. 130-134 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
General Class ◽  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Direct Generalization ◽  
Age Dependent ◽  
General Class Of Functions ◽  
Branching Process With Immigration ◽  
Necessary And Sufficient ◽  
Class Of Functions

Let {Z(t)}t0be an age-dependent branching process with immigration. For a general class of functions Φ(x), a necessary and sufficient condition is given for whenE{Φ (Z(t))} &lt;∞. This result is a direct generalization of a theorem proven for the branching process without immigration.

scholarly journals The weak law of large numbers for nonnegative summands

2018 ◽  
Vol 50 (A) ◽  
pp. 241-252
Author(s):  
Eugene Seneta
Keyword(s):  
Branching Process ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Sufficient Condition ◽  
Specific Function ◽  
Slowly Varying Function ◽  
Necessary And Sufficient Condition ◽  
Large Numbers ◽  
Norming Sequence ◽  
Necessary And Sufficient

Abstract Khintchine's (necessary and sufficient) slowly varying function condition for the weak law of large numbers (WLLN) for the sum of n nonnegative, independent and identically distributed random variables is used as an overarching (sufficient) condition for the case that the number of summands is more generally [cn],cn→∞. Either the norming sequence {an},an→∞, or the number of summands sequence {cn}, can be chosen arbitrarily. This theorem generalizes results from a motivating branching process setting in which Khintchine's sufficient condition is automatically satisfied. A second theorem shows that Khintchine's condition is necessary for the generalized WLLN when it holds with cn→∞ and an→∞. Theorem 3, which is known, gives a necessary and sufficient condition for Khintchine's WLLN to hold with cn=n and an a specific function of n; it is extended to general cn subject to a growth restriction in Theorem 4. Section 6 returns to the branching process setting.

On the identification of continuous-time Markov chains with a given invariant measure

1994 ◽  
Vol 31 (04) ◽  
pp. 897-910
Author(s):  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Continuous Time ◽  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Birth Process ◽  
Continuous Time Markov Chains ◽  
Branching Process With Immigration ◽  
Birth Processes ◽  
Necessary And Sufficient

In [14] a necessary and sufficient condition was obtained for there to exist uniquely a Q-process with a specified invariant measure, under the assumption that Q is a stable, conservative, single-exit matrix. The purpose of this note is to demonstrate that, for an arbitrary stable and conservative q-matrix, the same condition suffices for the existence of a suitable Q-process, but that this process might not be unique. A range of examples is considered, including pure-birth processes, a birth process with catastrophes, birth-death processes and the Markov branching process with immigration.

scholarly journals Fixation in Conditional Branching Process Models in Population Genetics

2007 ◽  
Vol 44 (4) ◽  
pp. 1103-1110 ◽  
Author(s):  
Thomas Prince ◽  
Neville Weber
Keyword(s):  
Population Genetics ◽  
Process Model ◽  
Branching Process ◽  
Process Models ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Alternative Version ◽  
Necessary And Sufficient ◽  
Insight Into

An alternative version of the necessary and sufficient condition for almost sure fixation in the conditional branching process model is derived. This formulation provides an insight into why the examples considered in Buckley and Seneta (1983) all have the same condition for fixation.

A note on the subcritical generalized age-dependent branching process

1976 ◽  
Vol 13 (04) ◽  
pp. 798-803
Author(s):  
R. A. Doney
Keyword(s):  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

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.

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