scholarly journals Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

2019 ◽  
Vol 23 ◽  
pp. 797-802
Author(s):  
Raphaël Cerf ◽  
Joseba Dalmau
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Branching Process ◽  
Invariant Probability Measure ◽  
Classical Formula ◽  
Multitype Branching Process ◽  
Primitive Matrix ◽  
Watson Process

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

1994 ◽  
Vol 26 (1) ◽  
pp. 80-103 ◽  
Author(s):  
Catherine Bouton ◽  
Gilles Pagès
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Convergence In Distribution ◽  
One Dimensional ◽  
Open Set ◽  
Recurrent Markov Chain ◽  
The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

A maximum sequence in a critical multitype branching process

1991 ◽  
Vol 28 (04) ◽  
pp. 893-897
Author(s):  
Aurel Spåtaru
Keyword(s):  
Spectral Radius ◽  
Branching Process ◽  
Multitype Branching Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
The Right ◽  
Maximum Sequence

Let (Zn ) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn ) consider the right eigenvector u = (u 1, · ··, up ) > 0, and set . It is shown that lim inf, lim sup whenever Z 0 = i, where .

Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

1994 ◽  
Vol 26 (01) ◽  
pp. 80-103 ◽  
Author(s):  
Catherine Bouton ◽  
Gilles Pagès
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Convergence In Distribution ◽  
One Dimensional ◽  
Open Set ◽  
Recurrent Markov Chain ◽  
The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

A maximum sequence in a critical multitype branching process

1991 ◽  
Vol 28 (4) ◽  
pp. 893-897 ◽  
Author(s):  
Aurel Spåtaru
Keyword(s):  
Spectral Radius ◽  
Branching Process ◽  
Multitype Branching Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
The Right ◽  
Maximum Sequence

Let (Zn) be a p-type positively regular and non-singular critical Galton–Watson process with finite second moments. Associated with the spectral radius 1 of the mean matrix of (Zn) consider the right eigenvector u = (u1, · ··, up) > 0, and set . It is shown that lim inf, lim sup whenever Z0 = i, where .

scholarly journals On the geometric growth in a class of homogeneous multitype Markov chain

2005 ◽  
Vol 42 (4) ◽  
pp. 1015-1030 ◽  
Author(s):  
M. González ◽  
R. Martínez ◽  
M. Mota
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Nonnegative Integer ◽  
Branching Process ◽  
Branching Processes ◽  
Multitype Branching Process ◽  
Supercritical Case ◽  
Theoretical Results ◽  
Geometric Growth

In this paper, we investigate the geometric growth of homogeneous multitype Markov chains whose states have nonnegative integer coordinates. Such models are considered in a situation similar to the supercritical case for branching processes. Finally, our general theoretical results are applied to a class of controlled multitype branching process in which the control is random.

scholarly journals On the geometric growth in a class of homogeneous multitype Markov chain

2005 ◽  
Vol 42 (04) ◽  
pp. 1015-1030 ◽  
Author(s):  
M. González ◽  
R. Martínez ◽  
M. Mota
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Nonnegative Integer ◽  
Branching Process ◽  
Branching Processes ◽  
Multitype Branching Process ◽  
Supercritical Case ◽  
Theoretical Results ◽  
Geometric Growth

In this paper, we investigate the geometric growth of homogeneous multitype Markov chains whose states have nonnegative integer coordinates. Such models are considered in a situation similar to the supercritical case for branching processes. Finally, our general theoretical results are applied to a class of controlled multitype branching process in which the control is random.

scholarly journals Demographic Dynamics in Multitype Populations with Migrations

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 246
Author(s):  
Manuel Molina-Fernández ◽  
Manuel Mota-Medina
Keyword(s):  
Mathematical Modeling ◽  
Population Dynamics ◽  
Branching Process ◽  
Probability Model ◽  
Biological Systems ◽  
Research Work ◽  
General Setting ◽  
Process Theory ◽  
Multitype Branching Process ◽  
Demographic Dynamics

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

scholarly journals Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 55
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Renewal Process ◽  
Markov Chain Model ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Chain Model ◽  
Markov Renewal Processes ◽  
Risky Bonds ◽  
Markov Structure

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

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