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 > 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 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

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.

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.

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

An indication of the asymptotic nature of the mendelian Markov process

1968 ◽  
Vol 5 (02) ◽  
pp. 350-356 ◽  
Author(s):  
R. G. Khazanie
Keyword(s):  
Markov Process ◽  
Transition Probabilities ◽  
Asymptotic Nature ◽  
Finite Markov Process ◽  
One Step ◽  
Step Transition ◽  
The One ◽  
Image Position

Consider a finite Markov process {Xn } described by the one-step transition probabilities In describing the transition probabilities in the above manner we are adopting the convention that (0)0 = 1 so that the states 0 and M are absorbing, and the states 1,2,···,M-1 are transient.

The departure process from the GI/G/1 Queue

1969 ◽  
Vol 6 (03) ◽  
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.

scholarly journals Markov Chain Models for the Stochastic Modeling of Pitting Corrosion

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
A. Valor ◽  
F. Caleyo ◽  
L. Alfonso ◽  
J. C. Velázquez ◽  
J. M. Hallen
Keyword(s):  
Markov Process ◽  
Pitting Corrosion ◽  
Probability Function ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Closed Form Solution ◽  
Form Solution ◽  
Extreme Value Statistics ◽  
Pit Depth ◽  
Transition Probability Function

The stochastic nature of pitting corrosion of metallic structures has been widely recognized. It is assumed that this kind of deterioration retains no memory of the past, so only the current state of the damage influences its future development. This characteristic allows pitting corrosion to be categorized as a Markov process. In this paper, two different models of pitting corrosion, developed using Markov chains, are presented. Firstly, a continuous-time, nonhomogeneous linear growth (pure birth) Markov process is used to model external pitting corrosion in underground pipelines. A closed-form solution of the system of Kolmogorov's forward equations is used to describe the transition probability function in a discrete pit depth space. The transition probability function is identified by correlating the stochastic pit depth mean with the empirical deterministic mean. In the second model, the distribution of maximum pit depths in a pitting experiment is successfully modeled after the combination of two stochastic processes: pit initiation and pit growth. Pit generation is modeled as a nonhomogeneous Poisson process, in which induction time is simulated as the realization of a Weibull process. Pit growth is simulated using a nonhomogeneous Markov process. An analytical solution of Kolmogorov's system of equations is also found for the transition probabilities from the first Markov state. Extreme value statistics is employed to find the distribution of maximum pit depths.

An indication of the asymptotic nature of the mendelian Markov process

1968 ◽  
Vol 5 (2) ◽  
pp. 350-356 ◽  
Author(s):  
R. G. Khazanie
Keyword(s):  
Markov Process ◽  
Transition Probabilities ◽  
Asymptotic Nature ◽  
Finite Markov Process ◽  
One Step ◽  
Step Transition ◽  
The One ◽  
Image Position

Consider a finite Markov process {Xn} described by the one-step transition probabilities In describing the transition probabilities in the above manner we are adopting the convention that (0)0 = 1 so that the states 0 and M are absorbing, and the states 1,2,···,M-1 are transient.

scholarly journals A MARKOV PROCESS OF GENE FREQUENCY CHANGE IN A GEOGRAPHICALLY STRUCTURED POPULATION

Genetics ◽  
1974 ◽  
Vol 76 (2) ◽  
pp. 367-377
Author(s):  
Takeo Maruyama
Keyword(s):  
Genetic Variation ◽  
Markov Process ◽  
Diffusion Process ◽  
Gene Frequency ◽  
Transition Probabilities ◽  
Large Population ◽  
Sample Path ◽  
Time Parameter ◽  
Frequency Change ◽  
Structured Population

ABSTRACT A Markov process (chain) of gene frequency change is derived for a geographically-structured model of a population. The population consists of colonies which are connected by migration. Selection operates in each colony independently. It is shown that there exists a stochastic clock that transforms the originally complicated process of gene frequency change to a random walk which is independent of the geographical structure of the population. The time parameter is a local random time that is dependent on the sample path. In fact, if the alleles are selectively neutral, the time parameter is exactly equal to the sum of the average local genetic variation appearing in the population, and otherwise they are approximately equal. The Kolmogorov forward and backward equations of the process are obtained. As a limit of large population size, a diffusion process is derived. The transition probabilities of the Markov chain and of the diffusion process are obtained explicitly. Certain quantities of biological interest are shown to be independent of the population structure. The quantities are the fixation probability of a mutant, the sum of the average local genetic variation and the variation summed over the generations in which the gene frequency in the whole population assumes a specified value.

scholarly journals Random Walks with Invariant Loop Probabilities: Stereographic Random Walks

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 729
Author(s):  
Miquel Montero
Keyword(s):  
Random Walks ◽  
Stereographic Projection ◽  
Transition Probabilities ◽  
Simple Random Walk ◽  
General Introduction ◽  
Elliptic Case ◽  
One Step ◽  
Underlying Geometry ◽  
Step Transition ◽  
Wide Family

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.

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