scholarly journals Transformations of Galton-Watson processes and linear fractional reproduction

2007 ◽  
Vol 39 (4) ◽  
pp. 1036-1053 ◽  
Author(s):  
F. C. Klebaner ◽  
U. Rösler ◽  
S. Sagitov
Keyword(s):  
Markov Chain ◽  
Time Reversal ◽  
Harmonic Functions ◽  
Invariant Measures ◽  
Graphical Representation ◽  
Simple Random Walk ◽  
Time Process ◽  
Offspring Distribution ◽  
Watson Process

By establishing general relationships between branching transformations (Harris-Sevastyanov, Lamperti-Ney, time reversals, and Asmussen-Sigman) and Markov chain transforms (Doob's h-transform, time reversal, and the cone dual), we discover a deeper connection between these transformations with harmonic functions and invariant measures for the process itself and its space-time process. We give a classification of the duals into Doob's h-transform, pathwise time reversal, and cone reversal. Explicit results are obtained for the linear fractional offspring distribution. Remarkably, for this case, all reversals turn out to be a Galton-Watson process with a dual reproduction law and eternal particle or some kind of immigration. In particular, we generalize a result of Klebaner and Sagitov (2002) in which only a geometric offspring distribution was considered. A new graphical representation in terms of an associated simple random walk on N2 allows for illuminating picture proofs of our main results concerning transformations of the linear fractional Galton-Watson process.

scholarly journals Transformations of Galton-Watson processes and linear fractional reproduction

2007 ◽  
Vol 39 (04) ◽  
pp. 1036-1053 ◽  
Author(s):  
F. C. Klebaner ◽  
U. Rösler ◽  
S. Sagitov
Keyword(s):  
Markov Chain ◽  
Time Reversal ◽  
Harmonic Functions ◽  
Invariant Measures ◽  
Graphical Representation ◽  
Simple Random Walk ◽  
Time Process ◽  
Offspring Distribution ◽  
Watson Process

By establishing general relationships between branching transformations (Harris-Sevastyanov, Lamperti-Ney, time reversals, and Asmussen-Sigman) and Markov chain transforms (Doob's h-transform, time reversal, and the cone dual), we discover a deeper connection between these transformations with harmonic functions and invariant measures for the process itself and its space-time process. We give a classification of the duals into Doob's h-transform, pathwise time reversal, and cone reversal. Explicit results are obtained for the linear fractional offspring distribution. Remarkably, for this case, all reversals turn out to be a Galton-Watson process with a dual reproduction law and eternal particle or some kind of immigration. In particular, we generalize a result of Klebaner and Sagitov (2002) in which only a geometric offspring distribution was considered. A new graphical representation in terms of an associated simple random walk on N 2 allows for illuminating picture proofs of our main results concerning transformations of the linear fractional Galton-Watson process.

A unifying approach to branching processes in a varying environment

2020 ◽  
Vol 57 (1) ◽  
pp. 196-220
Author(s):  
Götz Kersting
Keyword(s):  
Time Dependence ◽  
Population Size ◽  
Limit Distribution ◽  
Branching Processes ◽  
Type Result ◽  
Offspring Distribution ◽  
Watson Process ◽  
Square Integrability ◽  
Varying Environment

AbstractBranching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$, properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$, conditioned on the event $Z_n \gt 0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

Description and symmetry of aperiodic crystals

2018 ◽  
Author(s):  
Ted Janssen ◽  
Gervais Chapuis ◽  
Marc de Boissieu
Keyword(s):  
Time Reversal ◽  
Mathematical Concept ◽  
New Materials ◽  
Magnetic Systems ◽  
Atomic Structures ◽  
Modulated Phases ◽  
Quasiperiodic Functions ◽  
Aperiodic Crystals ◽  
Superspace Description

This chapter first introduces the mathematical concept of aperiodic and quasiperiodic functions, which will form the theoretical basis of the superspace description of the new recently discovered forms of matter. They are divided in three groups, namely modulated phases, composites, and quasicrystals. It is shown how the atomic structures and their symmetry can be characterized and described by the new concept. The classification of superspace groups is introduced along with some examples. For quasicrystals, the notion of approximants is also introduced for a better understanding of their structures. Finally, alternatives for the descriptions of the new materials are presented along with scaling symmetries. Magnetic systems and time-reversal symmetry are also introduced.

scholarly journals Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 179-196
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Obstacle Problem ◽  
Metric Spaces ◽  
Local Property ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Regular Boundary ◽  
Complete Metric Spaces ◽  
Regularity Results

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

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.

scholarly journals A new pipeline for structural characterization and classification of RNA-Seq microbiome data

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Sebastian Racedo ◽  
Ivan Portnoy ◽  
Jorge I. Vélez ◽  
Homero San-Juan-Vergara ◽  
Marco Sanjuan ◽  
...  
Keyword(s):  
Classification Accuracy ◽  
High Throughput Sequencing ◽  
Compositional Data ◽  
Graphical Representation ◽  
Synthetic Data ◽  
Gene Sequencing ◽  
Rrna Gene ◽  
Interaction Patterns ◽  
Simulation Experiments

Abstract Background High-throughput sequencing enables the analysis of the composition of numerous biological systems, such as microbial communities. The identification of dependencies within these systems requires the analysis and assimilation of the underlying interaction patterns between all the variables that make up that system. However, this task poses a challenge when considering the compositional nature of the data coming from DNA-sequencing experiments because traditional interaction metrics (e.g., correlation) produce unreliable results when analyzing relative fractions instead of absolute abundances. The compositionality-associated challenges extend to the classification task, as it usually involves the characterization of the interactions between the principal descriptive variables of the datasets. The classification of new samples/patients into binary categories corresponding to dissimilar biological settings or phenotypes (e.g., control and cases) could help researchers in the development of treatments/drugs. Results Here, we develop and exemplify a new approach, applicable to compositional data, for the classification of new samples into two groups with different biological settings. We propose a new metric to characterize and quantify the overall correlation structure deviation between these groups and a technique for dimensionality reduction to facilitate graphical representation. We conduct simulation experiments with synthetic data to assess the proposed method’s classification accuracy. Moreover, we illustrate the performance of the proposed approach using Operational Taxonomic Unit (OTU) count tables obtained through 16S rRNA gene sequencing data from two microbiota experiments. Also, compare our method’s performance with that of two state-of-the-art methods. Conclusions Simulation experiments show that our method achieves a classification accuracy equal to or greater than 98% when using synthetic data. Finally, our method outperforms the other classification methods with real datasets from gene sequencing experiments.

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.

Truncation approximations of invariant measures for Markov chains

1998 ◽  
Vol 35 (03) ◽  
pp. 517-536 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Transition Matrix ◽  
Invariant Measures ◽  
Invariant Distribution ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (01) ◽  
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 timegivenXn=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 forA(J) (J →∞) are given for these examples.

scholarly journals Design of a general brain-computer interface

2011 ◽  
Vol 22 (6) ◽  
pp. 638-646 ◽  
Author(s):  
Alessandro B. Benevides ◽  
Mário Sarcinelli-Filho ◽  
Teodiano F. Bastos Filho
Keyword(s):  
Feature Extraction ◽  
Brain Computer Interface ◽  
Sampling Frequency ◽  
Computer Interface ◽  
Eeg Signal ◽  
Time Process ◽  
Class Separation ◽  
Power Spectral ◽  
Mental Tasks

This paper presents the classification of three mental tasks, using the EEG signal and simulating a real-time process, what is known as pseudo-online technique. The Bayesian classifier is used to recognize the mental tasks, the feature extraction uses the Power Spectral Density, and the Sammon map is used to visualize the class separation. The choice of the EEG channel and sampling frequency is based on the Kullback-Leibler symmetric divergence and a reclassification model is proposed to stabilize the classifications.

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