Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Consider a continuous-time Markov process with transition rates matrixQin the state space Λ ⋃ {0}. In the associated Fleming-Viot processNparticles evolve independently in Λ with transition rates matrixQuntil one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When Λ is finite, we show that the empirical distribution of the particles at a fixed time converges asN→ ∞ to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process withNparticles converges asN→ ∞ to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1 /N.

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Dynamic performance evaluation of multi – state systems under non – homogeneous continuous time Markov process degradation using lifetimes in terms of order statistics

Proceedings of the Institution of Mechanical Engineers Part O Journal of Risk and Reliability ◽

10.1177/1748006x17695710 ◽

2017 ◽

Vol 231 (3) ◽

pp. 255-264 ◽

Cited By ~ 5

Author(s):

Funda Iscioglu

Keyword(s):

Markov Process ◽

Order Statistics ◽

Continuous Time ◽

Dynamic Performance ◽

Real Life ◽

State Change ◽

Transition Rates ◽

Lifetime Analysis ◽

Complete Failure ◽

Effect Of Age

In multi-state modelling a system and its components have a range of performance levels from perfect functioning to complete failure. Such a modelling is more flexible to understand the behaviour of mechanical systems. To evaluate a system’s dynamic performance, lifetime analysis of a multi-state system has been considered in many research articles. The order statistics related analysis for the lifetime properties of multi-state k-out-of-n systems have recently been studied in the literature in case of homogeneous continuous time Markov process assumption. In this paper, we develop the reliability measures for multi-state k-out-of-n systems by assuming a non-homogeneous continuous time Markov process for the components which provides time dependent transition rates between states of the components. Therefore, we capture the effect of age on the state change of the components in the analysis which is typical of many systems and more practical to use in real life applications.

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On certain unrestricted, linear, unit step, continuous time random walks

Journal of Applied Probability ◽

10.1017/s0021900200035385 ◽

1971 ◽

Vol 8 (02) ◽

pp. 374-380

Author(s):

A. E. Gibson ◽

B. W. Conolly

Keyword(s):

Random Walk ◽

Continuous Time ◽

Queueing Theory ◽

Basic Assumption ◽

The Other ◽

Time Intervals ◽

Queueing Process ◽

Unit Step ◽

Continuous Time Random Walks ◽

The One

A basic process of simple queueing theory is S(t), an integer valued stochastic process which represents the “number of clients present, including the one in service” at epoch t. The queueing context physically limits S(t) to non-negative values, but if this impenetrable barrier is removed thereby permitting S(t) to assume negative values as well, we have a “randomized random walk” in which S(t) represents the position at time t of a mythical particle which moves on the x-axis according to the rules of the walk. The nomenclature “randomized random walk” is due to Feller (1966a). S(t) changes by unit positive or negative amounts, and our basic assumption is that the time intervals τ/σ separating successive positive/negative steps are i.i.d. with continuous d.f.'s. A(t)/B(t) (A(0)/B(0) = 0) possessing p.d.f.'s. a(t)/b(t): τ and σ are further assumed to be statistically independent. When A(t) = 1 – e–λt and B(t) = 1 – e–μt , we have a generalization of the M/M/1 queueing process which we shall denote by M/M. The walk generated by A(t) = 1 – e–λt and B(t) arbitrary may similarly be denoted by M/G. For G/M we clearly mean A(t) arbitrary and B(t) = 1 – e–μt . M/M has received extensive treatment elsewhere (cf. Feller (1966a), (1966b); Takács (1967); Gibson (1968); Conolly (1971)), and will not be considered here. Our interest centers on certain aspects of M/G and G/M, but since each is the dual of the other (loosely, G/M is M/G “upside down”) it is sufficient to restrict attention to one of them, as convenient; pointing out, where necessary, the interpretation in terms of the other.

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Superposition of Interacting Aggregated Continuous-Time Markov Chains

Advances in Applied Probability ◽

10.1017/s0001867800027798 ◽

1997 ◽

Vol 29 (01) ◽

pp. 56-91

Author(s):

Frank Ball ◽

Robin K. Milne ◽

Ian D. Tame ◽

Geoffrey F. Yeo

Keyword(s):

Markov Chains ◽

Continuous Time ◽

Stochastic Modelling ◽

The Other ◽

Transition Rates ◽

Matrix Methods ◽

Joint Distributions ◽

Invariant Distributions ◽

Reliability Modelling ◽

Finite Markov Chains

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by a common partitioning of the state space. The interaction is assumed to arise from dependence of some of the transition rates for a given subsystem at a specified time on the states of the other subsystems at that time. With two subsystem classes, labelled 0 and 1, the superposition process arising from a system counts the number of subsystems in the latter class. Key structure and results from the theory of aggregated Markov processes are summarized. These are then applied also to superposition processes. In particular, we consider invariant distributions for the level m entry process, marginal and joint distributions for sojourn-times of the superposition process at its various levels, and moments and correlation functions associated with these distributions. The distributions are obtained mainly by using matrix methods, though an approach based on point process methods and conditional probability arguments is outlined. Conditions under which an interacting aggregated Markov chain is reversible are established. The ideas are illustrated with simple examples for which numerical results are obtained using Matlab. Motivation for this study has come from stochastic modelling of the behaviour of ion channels; another application is in reliability modelling.

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Embedded Markov chain approximations in Skorokhod topologies

Probability and Mathematical Statistics ◽

10.19195/0208-4147.39.2.2 ◽

2019 ◽

Vol 39 (2) ◽

pp. 259-277

Author(s):

Björn Böttcher

Keyword(s):

Markov Chains ◽

Continuous Time ◽

Waiting Times ◽

Linear Interpolation ◽

Step Function ◽

The Other ◽

Embedded Markov Chain ◽

Converse Statement ◽

The One ◽

Continuous Time Processes

We prove a J1-tightness condition for embedded Markov chains and discuss four Skorokhod topologies in a unified manner. To approximate a continuous time stochastic process by discrete time Markov chains, one has several options to embed the Markov chains into continuous time processes. On the one hand, there is a Markov embedding which uses exponential waiting times. On the other hand, each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for J1, the linear interpolation embedding forM1, the multistep embedding for J2 and a more general embedding for M2. We show that the convergence of the step function embedding in J1 implies the convergence of the other embeddings in the corresponding topologies. For the converse statement, a J1-tightness condition for embedded time-homogeneous Markov chains is given.Additionally, it is shown that J1 convergence is equivalent to the joint convergence in M1 and J2.

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Superposition of Interacting Aggregated Continuous-Time Markov Chains

Advances in Applied Probability ◽

10.2307/1427861 ◽

1997 ◽

Vol 29 (1) ◽

pp. 56-91 ◽

Cited By ~ 15

Author(s):

Frank Ball ◽

Robin K. Milne ◽

Ian D. Tame ◽

Geoffrey F. Yeo

Keyword(s):

Markov Chains ◽

Continuous Time ◽

Stochastic Modelling ◽

The Other ◽

Transition Rates ◽

Matrix Methods ◽

Joint Distributions ◽

Invariant Distributions ◽

Reliability Modelling ◽

Finite Markov Chains

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by a common partitioning of the state space. The interaction is assumed to arise from dependence of some of the transition rates for a given subsystem at a specified time on the states of the other subsystems at that time. With two subsystem classes, labelled 0 and 1, the superposition process arising from a system counts the number of subsystems in the latter class. Key structure and results from the theory of aggregated Markov processes are summarized. These are then applied also to superposition processes. In particular, we consider invariant distributions for the level m entry process, marginal and joint distributions for sojourn-times of the superposition process at its various levels, and moments and correlation functions associated with these distributions. The distributions are obtained mainly by using matrix methods, though an approach based on point process methods and conditional probability arguments is outlined. Conditions under which an interacting aggregated Markov chain is reversible are established. The ideas are illustrated with simple examples for which numerical results are obtained using Matlab. Motivation for this study has come from stochastic modelling of the behaviour of ion channels; another application is in reliability modelling.

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Monetary risk measures for stochastic processes via Orlicz duality

Decisions in Economics and Finance ◽

10.1007/s10203-021-00334-x ◽

2021 ◽

Author(s):

Christos E. Kountzakis ◽

Damiano Rossello

Keyword(s):

Financial Performance ◽

Stochastic Processes ◽

Continuous Time ◽

Risk Measures ◽

Cash Flows ◽

The Other ◽

Future Evolution ◽

Heavy Tailed Distributions ◽

Heavy Tailed ◽

The One

AbstractIn this article, we extend the framework of monetary risk measures for stochastic processes to account for heavy tailed distributions of random cash flows evolving over a fixed trading horizon. To this end, we transfer the $$L^p$$ L p -duality underlying the representation of monetary risk measures to a more flexible Orlicz duality, in spaces of stochastic processes modelling random future evolution of financial values in continuous time over a finite horizon. This contributes, on the one hand, to the theory of real-valued monetary risk measures for processes and, on the other hand, supports a new representation of acceptability indices of financial performance.

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On certain unrestricted, linear, unit step, continuous time random walks

Journal of Applied Probability ◽

10.2307/3211907 ◽

1971 ◽

Vol 8 (2) ◽

pp. 374-380

Author(s):

A. E. Gibson ◽

B. W. Conolly

Keyword(s):

Random Walk ◽

Continuous Time ◽

Queueing Theory ◽

Basic Assumption ◽

The Other ◽

Time Intervals ◽

Queueing Process ◽

Unit Step ◽

Continuous Time Random Walks ◽

The One

A basic process of simple queueing theory is S(t), an integer valued stochastic process which represents the “number of clients present, including the one in service” at epoch t. The queueing context physically limits S(t) to non-negative values, but if this impenetrable barrier is removed thereby permitting S(t) to assume negative values as well, we have a “randomized random walk” in which S(t) represents the position at time t of a mythical particle which moves on the x-axis according to the rules of the walk. The nomenclature “randomized random walk” is due to Feller (1966a). S(t) changes by unit positive or negative amounts, and our basic assumption is that the time intervals τ/σ separating successive positive/negative steps are i.i.d. with continuous d.f.'s. A(t)/B(t) (A(0)/B(0) = 0) possessing p.d.f.'s. a(t)/b(t): τ and σ are further assumed to be statistically independent. When A(t) = 1 – e–λt and B(t) = 1 – e–μt, we have a generalization of the M/M/1 queueing process which we shall denote by M/M. The walk generated by A(t) = 1 – e–λt and B(t) arbitrary may similarly be denoted by M/G. For G/M we clearly mean A(t) arbitrary and B(t) = 1 – e–μt. M/M has received extensive treatment elsewhere (cf. Feller (1966a), (1966b); Takács (1967); Gibson (1968); Conolly (1971)), and will not be considered here. Our interest centers on certain aspects of M/G and G/M, but since each is the dual of the other (loosely, G/M is M/G “upside down”) it is sufficient to restrict attention to one of them, as convenient; pointing out, where necessary, the interpretation in terms of the other.

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Note on Selection of Coordinate Systems for Geodynamics

International Astronomical Union Colloquium ◽

10.1017/s0252921100051174 ◽

1975 ◽

Vol 26 ◽

pp. 395-407

Author(s):

S. Henriksen

Keyword(s):

Sea Level ◽

The Other ◽

Coordinate Systems ◽

Mean Sea Level ◽

The Earth ◽

The One ◽

Static Systems ◽

Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

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Stories and the Self

Journal of Media Psychology Theories Methods and Applications ◽

10.1027/1864-1105/a000255 ◽

2020 ◽

Vol 32 (2) ◽

pp. 47-58 ◽

Cited By ~ 1

Author(s):

Stefan Krause ◽

Markus Appel

Keyword(s):

Meta Analysis ◽

The Self ◽

The Other ◽

Lower Degree ◽

Contrast Effects ◽

Self Perception ◽

Self Perceptions ◽

Other Hand ◽

The One ◽

Implicit And Explicit

Abstract. Two experiments examined the influence of stories on recipients’ self-perceptions. Extending prior theory and research, our focus was on assimilation effects (i.e., changes in self-perception in line with a protagonist’s traits) as well as on contrast effects (i.e., changes in self-perception in contrast to a protagonist’s traits). In Experiment 1 ( N = 113), implicit and explicit conscientiousness were assessed after participants read a story about either a diligent or a negligent student. Moderation analyses showed that highly transported participants and participants with lower counterarguing scores assimilate the depicted traits of a story protagonist, as indicated by explicit, self-reported conscientiousness ratings. Participants, who were more critical toward a story (i.e., higher counterarguing) and with a lower degree of transportation, showed contrast effects. In Experiment 2 ( N = 103), we manipulated transportation and counterarguing, but we could not identify an effect on participants’ self-ascribed level of conscientiousness. A mini meta-analysis across both experiments revealed significant positive overall associations between transportation and counterarguing on the one hand and story-consistent self-reported conscientiousness on the other hand.

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