A theorem on compositions of random probability generating functions and applications to branching processes with random environments

1972 ◽  
Vol 9 (01) ◽  
pp. 1-12 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Generating Functions ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Environments ◽  
Probability Generating Functions ◽  
Random Probability

The results of this paper deal directly with the behavior of compositions of random probability generating functions. These results are then applied to the theory of branching processes with random environments. In particular, it is shown that it is not possible to strengthen many of the existing limit theorems on this subject.

A theorem on compositions of random probability generating functions and applications to branching processes with random environments

1972 ◽  
Vol 9 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Generating Functions ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Environments ◽  
Probability Generating Functions ◽  
Random Probability

The results of this paper deal directly with the behavior of compositions of random probability generating functions. These results are then applied to the theory of branching processes with random environments. In particular, it is shown that it is not possible to strengthen many of the existing limit theorems on this subject.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

Branching processes with varying and random geometric offspring distributions

1975 ◽  
Vol 12 (01) ◽  
pp. 135-141 ◽  
Author(s):  
Niels Keiding ◽  
John E. Nielsen
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Conditional Expectation ◽  
Elementary Proof ◽  
Branching Processes ◽  
Random Environments ◽  
Subcritical Case ◽  
Supercritical Case ◽  
Extinction Probabilities ◽  
Varying Environments

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.

Generalizations of the elementary renewal theorem to distributions defined by concave recurrence relations

1977 ◽  
Vol 14 (3) ◽  
pp. 516-526
Author(s):  
W. Reh
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Probability Distributions ◽  
Branching Processes ◽  
General Procedure ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Probability Generating Functions ◽  
Dependent Probability

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.

scholarly journals Branching processes in random environment die slowly

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Vladimir Vatutin ◽  
Andreas Kyprianou
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Generating Functions ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Limit Distributions ◽  
Conditional Limit Theorems ◽  
International Audience ◽  
Conditional Limit

International audience Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.

GENERATING FUNCTIONS FOR STOCHASTIC CONTEXT-FREE GRAMMARS

1990 ◽  
Vol 04 (04) ◽  
pp. 553-572 ◽  
Author(s):  
MICHAEL G. THOMASON
Keyword(s):  
Generating Functions ◽  
Pattern Analysis ◽  
Branching Processes ◽  
Single Type ◽  
Probability Generating Functions ◽  
Syntactic Pattern ◽  
Multitype Branching Processes ◽  
Stochastic Context Free Grammars ◽  
Context Free ◽  
Context Free Grammars

Stochastic context-free grammars are an important tool in syntactic pattern analysis and other applications as well. This paper discusses major results in single-type and multitype branching processes used to study a grammar’s stochastic derivations. Probability generating functions are well-established as a tool in this area and are used extensively here.

Generalizations of the elementary renewal theorem to distributions defined by concave recurrence relations

1977 ◽  
Vol 14 (03) ◽  
pp. 516-526
Author(s):  
W. Reh
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Probability Distributions ◽  
Branching Processes ◽  
General Procedure ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Probability Generating Functions ◽  
Dependent Probability

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (3) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn} whose state space is the non-negative integers and whose transition probability matrix ║Pij║ possesses the representation where {Pr}, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z0 = k, a finite positive integer.

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