The Galton-Watson process with mean one

1967 ◽  
Vol 4 (3) ◽  
pp. 489-495 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Markov Chain ◽  
Transition Probabilities ◽  
Watson Process ◽  
Step Transition ◽  
Image Position ◽  
Number Of Individuals

Let Zn be the number of individuals in the n-th generation of a simple (discrete, one-type) Galton-Watson process, descended from a single ancestor. Write and assume, as usual, that 0 < F(0) < 1. It is well known that the p.g.f. of Zn is Fn(s), the n-th functional iterate of F(s), and that if the process is construed as a Markov chain on the states 0, 1, 2,…, then its n-step transition probabilities are given by

The Galton-Watson process with mean one

1967 ◽  
Vol 4 (03) ◽  
pp. 489-495 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Markov Chain ◽  
Transition Probabilities ◽  
Watson Process ◽  
Step Transition ◽  
Image Position ◽  
Number Of Individuals

Let Zn be the number of individuals in the n-th generation of a simple (discrete, one-type) Galton-Watson process, descended from a single ancestor. Write and assume, as usual, that 0 &lt; F(0) &lt; 1. It is well known that the p.g.f. of Zn is Fn (s), the n-th functional iterate of F(s), and that if the process is construed as a Markov chain on the states 0, 1, 2,…, then its n-step transition probabilities are given by

On non-dissipative Markov chains

1957 ◽  
Vol 53 (4) ◽  
pp. 825-835 ◽  
Author(s):  
J. G. Mauldon
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Usual Convention ◽  
One Step ◽  
Step Transition ◽  
Image Position

Consider a Markov chain with an enumerable infinity of states, labelled 0, 1, 2, …, whose one-step transition probabilities pij are independent of time. ThenI write and, departing slightly from the usual convention,Then it is known ((1), pp. 324–34, or (6)) that the limits πij always exist, and that

The reverse Galton-Watson process

1975 ◽  
Vol 12 (03) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn (w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn , which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN –1 &gt; 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ 2 &lt; ∞. Two related results about Zcn, 0 &lt; c &lt; 1, for Galton-Watson processes conclude the paper.

Single-shelf library-type Markov chains with infinitely many books

1977 ◽  
Vol 14 (02) ◽  
pp. 298-308 ◽  
Author(s):  
Peter R. Nelson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Natural Order ◽  
Order Book ◽  
Exit Boundary ◽  
One Step ◽  
Step Transition ◽  
The Right ◽  
Positive Recurrent

In a single-shelf library having infinitely many books B 1 , B 2 , …, the probability of selecting each book is assumed known. Books are removed one at a time and replaced in position k prior to the next removal. Books are moved either to the right or the left as is necessary to vacate position k. Those arrangements of books where after some finite position all the books are in natural order (book i occupies position i) are considered as states in an infinite Markov chain. When k &gt; 1, we show that the chain can never be positive recurrent. When k = 1, we find the limits of ratios of one-step transition probabilities; and when k = 1 and the chain is transient, we find the Martin exit boundary.

The reverse Galton-Watson process

1975 ◽  
Vol 12 (3) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn(w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN–1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

scholarly journals A Diffusion Limit for Generalized Correlated Random Walks

2006 ◽  
Vol 43 (01) ◽  
pp. 60-73 ◽  
Author(s):  
Urs Gruber ◽  
Martin Schweizer
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Partial Sums ◽  
Diffusion Limit ◽  
Correlated Random Walk ◽  
Binomial Trees ◽  
Correlated Random Walks ◽  
Limiting Behaviour ◽  
Image Position

A generalized correlated random walk is a process of partial sums such that (X, Y) forms a Markov chain. For a sequence (X n ) of such processes in which each takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Y n . Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.

scholarly journals On the Convergence Rate of Bonus-Malus Systems

1992 ◽  
Vol 22 (2) ◽  
pp. 217-223 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Convergence Rate ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Limit Probability ◽  
Step Transition

AbstractUnder certain conditions, a Bonus-Malus system can be interpreted as a Markov chain whose n-step transition probabilities converge to a limit probability distribution. In this paper, the rate of the convergence is studied by means of the eigenvalues of the transition probability matrix of the Markov chain.

On a class of non-homogeneous Markov chains

1982 ◽  
Vol 92 (3) ◽  
pp. 527-534 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Homogeneous Markov Chain ◽  
Image Position

AbstractSuppose that {Xn} is a countable non-homogeneous Markov chain andIf converges for any i, l, m, j with , thenwhenever lim , whereas if converges, thenwhere and . The behaviour of transition probabilities between various groups of states is studied and criteria for recurrence and transience are given.

Some explicit results for correlated random walks

1989 ◽  
Vol 26 (4) ◽  
pp. 757-766 ◽  
Author(s):  
Ram Lal ◽  
U. Narayan Bhat
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Equilibrium Distribution ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Correlated Random Walks ◽  
Finite State ◽  
Direction Of Movement ◽  
Step Transition

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).

The departure process from the GI/G/1 Queue

1969 ◽  
Vol 6 (3) ◽  
pp. 704-707 ◽  
Author(s):  
Thomas L. Vlach ◽  
Ralph L. Disney
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Transition Probabilities ◽  
Two Dimensional ◽  
Departure Process ◽  
Dimensional State Space ◽  
Semi Markov Process ◽  
One Step ◽  
Step Transition ◽  
The One

The departure process from the GI/G/1 queue is shown to be a semi-Markov process imbedded at departure points with a two-dimensional state space. Transition probabilities for this process are defined and derived from the distributions of the arrival and service processes. The one step transition probabilities and a stationary distribution are obtained for the imbedded two-dimensional Markov chain.

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