Countable non-homogeneous Markov chains: asymptotic behaviour

1977 ◽  
Vol 9 (3) ◽  
pp. 542-552 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Asymptotic Properties ◽  
Homogeneous Markov Chain ◽  
Homogeneous Case ◽  
Main Concept ◽  
State Classification ◽  
Ratio Limit

The paper deals with asymptotic properties of the transition probabilities of a countable non-homogeneous Markov chain, the main concept used in the proofs being that of the tail σ-field of the chain. A state classification similar to that existing in the homogeneous case is given and a strong ratio limit property is shown to parallel the basic limit theorem for positive homogeneous chains. Some global asymptotic properties for null chains are also derived.

Countable non-homogeneous Markov chains: asymptotic behaviour

1977 ◽  
Vol 9 (03) ◽  
pp. 542-552 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Asymptotic Properties ◽  
Homogeneous Markov Chain ◽  
Homogeneous Case ◽  
Main Concept ◽  
State Classification ◽  
Ratio Limit

The paper deals with asymptotic properties of the transition probabilities of a countable non-homogeneous Markov chain, the main concept used in the proofs being that of the tail σ-field of the chain. A state classification similar to that existing in the homogeneous case is given and a strong ratio limit property is shown to parallel the basic limit theorem for positive homogeneous chains. Some global asymptotic properties for null chains are also derived.

Finite non-homogeneous Markov chains: Asymptotic behaviour

1976 ◽  
Vol 8 (3) ◽  
pp. 502-516 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Homogeneous Markov Chain ◽  
Stochastic Matrices ◽  
Weak Ergodicity ◽  
General Picture ◽  
Markov Chain Theory ◽  
Chain Theory ◽  
Probabilistic Nature

The paper is concerned with aspects of the behaviour of the products of finite stochastic matrices, the methods used in the proofs being of a probabilistic nature. The main result of the paper (Theorem 1) presents a general picture of the asymptotic behaviour of the transition probabilities between various groups of states. A unified treatment of some results of non-homogeneous Markov chain theory pertaining to weak ergodicity is then given.

Finite non-homogeneous Markov chains: Asymptotic behaviour

1976 ◽  
Vol 8 (03) ◽  
pp. 502-516 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Homogeneous Markov Chain ◽  
Stochastic Matrices ◽  
Weak Ergodicity ◽  
General Picture ◽  
Markov Chain Theory ◽  
Chain Theory ◽  
Probabilistic Nature

The paper is concerned with aspects of the behaviour of the products of finite stochastic matrices, the methods used in the proofs being of a probabilistic nature. The main result of the paper (Theorem 1) presents a general picture of the asymptotic behaviour of the transition probabilities between various groups of states. A unified treatment of some results of non-homogeneous Markov chain theory pertaining to weak ergodicity is then given.

scholarly journals The local limit theorem for sums of random variables defined on a homogeneous Markov chain in the case of the stable limit distribution law

1961 ◽  
Vol 1 (1-2) ◽  
pp. 7-16
Author(s):  
A. Aleškevičienė
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Distribution ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Homogeneous Markov Chain ◽  
Stable Limit ◽  
Distribution Law ◽  
Sums Of Random Variables ◽  
Stable Limit Distribution

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: A. Алешкявичене. Локальная предельная теорема для сумм случайных величин, связанных в однородную цепь Маркова в случае устойчивого предельного распределения A. Aleškevičienė. Lokalinė ribinė teorema atsitiktinių dydžių, surištų homogenine Markovo grandine, sumoms stabilaus ribinio dėsnio atveju  

An adaptive Markov Chain approach for probabilistic forecasting of categorical events

2020 ◽  
pp. 1-31
Author(s):  
Edward C.D. Pope ◽  
David B. Stephenson ◽  
David R. Jackson
Keyword(s):  
Markov Chain ◽  
Space Weather ◽  
Weather Forecasting ◽  
Transition Probabilities ◽  
Parametric Estimation ◽  
Lead Times ◽  
Homogeneous Markov Chain ◽  
Probabilistic Prediction ◽  
Geomagnetic Disturbances ◽  
Memory Parameter

Abstract Categorical probabilistic prediction is widely used for terrestrial and space weather forecasting as well as for other environmental forecasts. One example is a warning system for geomagnetic disturbances caused by space weather, which are often classified on a 10-level scale. The simplest approach assumes that the transition probabilities are stationary in time – the Homogeneous Markov Chain (HMC). We extend this approach by developing a flexible Non-Homogeneous Markov Chain (NHMC) model using Bayesian non-parametric estimation to describe the time-varying transition probabilities. The transition probabilities are updated using a modified Bayes’ rule that gradually forgets transitions in the distant past, with a tunable memory parameter. The approaches were tested by making daily geomagnetic state forecasts at lead times of 1-4 days and verified over the period 2000-2019 using the Rank Probability Score (RPS). Both HMC and NHMC models were found to be skilful at all lead times when compared with climatological forecasts. The NHMC forecasts with an optimal memory parameter of ~100 days were found to be substantially more skilful than the HMC forecasts, with an RPS skill for the NHMC of 10.5% and 5.6% for lead times of 1 and 4 days ahead, respectively. The NHMC is thus a viable alternative approach for forecasting geomagnetic disturbances, and could provide a new benchmark for producing operational forecasts. The approach is generic and applicable to other forecasts including discrete weather regimes or hydrological conditions, e.g. wet and dry days.

On a class of non-homogeneous Markov chains

1982 ◽  
Vol 92 (3) ◽  
pp. 527-534 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Homogeneous Markov Chain ◽  
Image Position

AbstractSuppose that {Xn} is a countable non-homogeneous Markov chain andIf converges for any i, l, m, j with , thenwhenever lim , whereas if converges, thenwhere and . The behaviour of transition probabilities between various groups of states is studied and criteria for recurrence and transience are given.

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