Wandering phenomena in infinite-allelic diffusion models

1982 ◽  
Vol 14 (3) ◽  
pp. 457-483 ◽  
Author(s):  
Tokuzo Shiga
Keyword(s):  
Population Genetics ◽  
Ergodic Theorem ◽  
Stationary Distribution ◽  
Diffusion Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
Individual Ergodic Theorem ◽  
Necessary And Sufficient

We introduce a class of infinite-dimensional diffusion processes which contains a limiting version of the Ohta–Kimura model in population genetics. For this a necessary and sufficient condition for existence of stationary distributions is obtained. We are especially interested in the case where there is no stationary distribution. Then it is shown that an individual ergodic theorem holds for a suitably centralized process. As a corollary the wandering distribution exists.

Wandering phenomena in infinite-allelic diffusion models

1982 ◽  
Vol 14 (03) ◽  
pp. 457-483 ◽  
Author(s):  
Tokuzo Shiga
Keyword(s):  
Population Genetics ◽  
Ergodic Theorem ◽  
Stationary Distribution ◽  
Diffusion Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
Individual Ergodic Theorem ◽  
Necessary And Sufficient

We introduce a class of infinite-dimensional diffusion processes which contains a limiting version of the Ohta–Kimura model in population genetics. For this a necessary and sufficient condition for existence of stationary distributions is obtained. We are especially interested in the case where there is no stationary distribution. Then it is shown that an individual ergodic theorem holds for a suitably centralized process. As a corollary the wandering distribution exists.

On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains

1993 ◽  
Vol 25 (01) ◽  
pp. 82-102
Author(s):  
M. G. Nair ◽  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Absorbing State ◽  
Necessary And Sufficient ◽  
Forward Equations ◽  
Quasi Stationary Distribution

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992). In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.

scholarly journals Characterizations of majorization on summable sequences

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2193-2202
Author(s):  
Kosuru Raju ◽  
Subhajit Saha
Keyword(s):  
Sequence Space ◽  
Dimensional Space ◽  
Sufficient Condition ◽  
Infinite Dimensional Space ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Summable Sequence ◽  
Necessary And Sufficient

In this paper, we prove a necessary and sufficient condition for majorization on the summable sequence space. For this we redefine the notion of majorization on an infinite dimensional space and therein investigate properties of the majorization. We also prove the infinite dimensional Schur-Horn type and Hardy-Littlewood-P?lya type theorems.

scholarly journals The point spectrum and residual spectrum of upper triangular operator matrices

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1759-1771
Author(s):  
Xiufeng Wu ◽  
Junjie Huang ◽  
Alatancang Chen
Keyword(s):  
Hilbert Spaces ◽  
Point Spectrum ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Residual Spectrum ◽  
Infinite Dimensional ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient

The point and residual spectra of an operator are, respectively, split into 1,2-point spectrum and 1,2-residual spectrum, based on the denseness and closedness of its range. Let H,K be infinite dimensional complex separable Hilbert spaces and write MX = (AX0B) ? B(H?K). For given operators A ? B(H) and B ? B(K), the sets ? X?B(K,H) ?+,i(MX)(+ = p,r;i = 1,2), are characterized. Moreover, we obtain some necessary and sufficient condition such that ?*,i(MX) = ?*,i(A) ?*,i(B) (* = p,r;i = 1,2) for every X ? B(K,H).

scholarly journals Fixation in Conditional Branching Process Models in Population Genetics

2007 ◽  
Vol 44 (4) ◽  
pp. 1103-1110 ◽  
Author(s):  
Thomas Prince ◽  
Neville Weber
Keyword(s):  
Population Genetics ◽  
Process Model ◽  
Branching Process ◽  
Process Models ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Alternative Version ◽  
Necessary And Sufficient ◽  
Insight Into

An alternative version of the necessary and sufficient condition for almost sure fixation in the conditional branching process model is derived. This formulation provides an insight into why the examples considered in Buckley and Seneta (1983) all have the same condition for fixation.

scholarly journals A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

2007 ◽  
Vol 44 (04) ◽  
pp. 938-949 ◽  
Author(s):  
Shui Feng ◽  
Feng-Yu Wang
Keyword(s):  
Population Genetics ◽  
Sobolev Inequality ◽  
Diffusion Processes ◽  
Exponential Convergence ◽  
Sobolev Inequalities ◽  
Abstract Space ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
Reasonable Alternative

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := { x ∈ [0, 1] N : ∑ i≥1 x i = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

scholarly journals Sequences and bases in p-Banach spaces

1986 ◽  
Vol 34 (1) ◽  
pp. 87-92
Author(s):  
M. A. Ariño
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dimensional Subspace ◽  
Basic Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Subspace

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.

On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains

1993 ◽  
Vol 25 (1) ◽  
pp. 82-102 ◽  
Author(s):  
M. G. Nair ◽  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Absorbing State ◽  
Necessary And Sufficient ◽  
Forward Equations ◽  
Quasi Stationary Distribution

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992).In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.

MULTIPLICATIVE RENORMALIZABILITY AND THE LAPLACE-BELTRAMI OPERATOR

1991 ◽  
Vol 06 (06) ◽  
pp. 955-976
Author(s):  
D. OLIVIER ◽  
G. VALENT
Keyword(s):  
Irreducible Representation ◽  
Isometry Group ◽  
Zero Mode ◽  
Beltrami Operator ◽  
Sufficient Condition ◽  
Representation Space ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Multiplicative Renormalizability

For some rank 1 non-linear σ models we prove that a necessary and sufficient condition of multiplicative renormalizability for composite fields is that they should be eigenfunctions of the coset Laplace-Beltrami operator. These eigenfunctions span the irreducible representation space of the isometry group and may be finite- or infinite-dimensional. The zero mode of the Laplace-Beltrami operator plays a particular role since it is not renormalized at all.

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