Some limit theorems for positive recurrent branching Markov chains: II

1998 ◽  
Vol 30 (03) ◽  
pp. 711-722 ◽  
Author(s):  
Krishna B. Athreya ◽  
Hye-Jeong Kang
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
General State ◽  
Position Distribution ◽  
Large Numbers ◽  
Recurrent Markov Chain ◽  
General State Space ◽  
Watson Process ◽  
Positive Recurrent

In this paper we consider a Galton-Watson process in which particles move according to a positive recurrent Markov chain on a general state space. We prove a law of large numbers for the empirical position distribution and also discuss the rate of this convergence.

Some limit theorems for positive recurrent branching Markov chains: II

1998 ◽  
Vol 30 (3) ◽  
pp. 711-722 ◽  
Author(s):  
Krishna B. Athreya ◽  
Hye-Jeong Kang
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
General State ◽  
Position Distribution ◽  
Large Numbers ◽  
Recurrent Markov Chain ◽  
General State Space ◽  
Watson Process ◽  
Positive Recurrent

In this paper we consider a Galton-Watson process in which particles move according to a positive recurrent Markov chain on a general state space. We prove a law of large numbers for the empirical position distribution and also discuss the rate of this convergence.

Some limit theorems for positive recurrent branching Markov chains: I

1998 ◽  
Vol 30 (03) ◽  
pp. 693-710 ◽  
Author(s):  
Krishna B. Athreya ◽  
Hye-Jeong Kang
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Position Distribution ◽  
Discrete State ◽  
Large Numbers ◽  
Watson Process ◽  
Positive Recurrent ◽  
Discrete State Space

In this paper we consider a Galton-Watson process whose particles move according to a Markov chain with discrete state space. The Markov chain is assumed to be positive recurrent. We prove a law of large numbers for the empirical position distribution and also discuss the large deviation aspects of this convergence.

Some limit theorems for positive recurrent branching Markov chains: I

1998 ◽  
Vol 30 (3) ◽  
pp. 693-710 ◽  
Author(s):  
Krishna B. Athreya ◽  
Hye-Jeong Kang
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Position Distribution ◽  
Discrete State ◽  
Large Numbers ◽  
Watson Process ◽  
Positive Recurrent ◽  
Discrete State Space

In this paper we consider a Galton-Watson process whose particles move according to a Markov chain with discrete state space. The Markov chain is assumed to be positive recurrent. We prove a law of large numbers for the empirical position distribution and also discuss the large deviation aspects of this convergence.

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (3) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (03) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

Semi-Markov processes on a general state space: α-theory and quasi-stationarity

1980 ◽  
Vol 30 (2) ◽  
pp. 187-200 ◽  
Author(s):  
E. Arjas ◽  
E. Nummelin ◽  
R. L. Tweedie
Keyword(s):  
State Space ◽  
Markov Processes ◽  
Limit Theorems ◽  
General State ◽  
General State Space ◽  
Quasistationary Distributions ◽  
Positive Recurrent ◽  
Semi Markov Processes

AbstractBy amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an α-theory is developed for general semi-Markov processes. It is shown that α-transient, α-recurrent and α-positive recurrent processes can be defined, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for α-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).

scholarly journals Limit theorems for multi-type general branching processes with population dependence

2020 ◽  
Vol 52 (4) ◽  
pp. 1127-1163
Author(s):  
Jie Yen Fan ◽  
Kais Hamza ◽  
Peter Jagers ◽  
Fima C. Klebaner
Keyword(s):  
Carrying Capacity ◽  
Population Model ◽  
General Framework ◽  
Limit Theorems ◽  
Mating Systems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Special Section ◽  
Large Numbers ◽  
Type Population

AbstractA general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

Sign in / Sign up

Export Citation Format

Share Document