On an urn problem of Paul and Tatiana Ehrenfest

1979 ◽  
Vol 86 (1) ◽  
pp. 127-130 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Transition Probabilities ◽  
The Other ◽  
Homogeneous Markov Chain ◽  
Apparent Discrepancy ◽  
Urn Model ◽  
Image Position

A short solution is given for the urn problem proposed by Paul and Tatiana Ehrenfest in 1907.In 1907 P. and T. Ehrenfest(3) proposed an urn model for the resolution of the apparent discrepancy between irreversibility and recurrence in Boltzmann's theory of gases (2). In this model it is assumed that m balls numbered 1, 2, …, m are distributed in two boxes. We perform a series of trials. In each trial we choose a number at random among 1, 2, …, m in such a way that each number has probability 1/m. If we choose j, then we transfer the ball numbered j from one box to the other. Denote by ξn the number of balls in the first box at the end of the nth trial. Initially there are ξ0 balls in the first box. If the trials are independent, then the sequence {ξn;n = 0, 1, 2,…} forms a homogeneous Markov chain with state space I = {0, 1, 2,…, m} and transition probabilities pi,i+1 = (m − i)/m for i = 0, l,…, m − 1, Pi,i−1 = i/m for i = 1, 2,…, m, and pi,k = 0 otherwise. The problem is to determine the transition probabilitiesfor i∈I, k∈I and n = 0,1, 2,….

ON ASTUMIAN'S PARADOX

2005 ◽  
Vol 05 (01) ◽  
pp. L109-L125
Author(s):  
EHRHARD BEHRENDS
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Transition Probabilities ◽  
Natural Generalization ◽  
The Other ◽  
Finite Markov Chain ◽  
Fair Coin ◽  
Winning Probability ◽  
Starting Position ◽  
Absorbing States

An Astumian game is defined by a finite Markov chain with state space S with precisely two absorbing states, the winning and the losing state. The other states are transient, one of them is the starting position. The game is said to be losing (respectively fair respectively winning) if the probability to be absorbed at the winning state is smaller than 0.5 (respectively = 0.5 respectively larger than 0.5). Astumian's paradox states that there are losing games on the same state space S a stochastic mixture of which is winning. (By "stochastic mixture" we mean that in each step one decides with the help of a fair coin whether to use the transition probabilities of the first or the second game.) Most of our results concern fair games. Mixtures are systematically investigated. Rather surprisingly, the winning probability of the mixture of fair games can be arbitrarily close to zero (or to one). Even more counter-intuitive are examples of definitely losing games (this means that the winning probability is exactly zero) such that the winning probability of the mixture is arbitrarily close to one. We show, however, that such extreme examples are possible only if one tolerates huge running times of the game. As a natural generalization one can also consider arbitrary mixtures: the fair coin is replaced by a biased one, with probability λ respectively 1-λ one plays with the first respectively the second game. It turns out that fair games exist such that — depending on the choice of λ — the λ-mixture can be fair, losing or winning.

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.

On sums of random variables defined on a two-state Markov chain

1970 ◽  
Vol 7 (3) ◽  
pp. 761-765 ◽  
Author(s):  
H. J. Helgert
Keyword(s):  
Markov Chain ◽  
Transition Probabilities ◽  
Random Variables ◽  
Homogeneous Markov Chain ◽  
Sums Of Random Variables ◽  
Image Position

Assume the sequence of random variables x0, x1, x2, ··· forms a two-state, homogeneous Markov chain with transition probabilities and initial probabilities

On sums of random variables defined on a two-state Markov chain

1970 ◽  
Vol 7 (03) ◽  
pp. 761-765 ◽  
Author(s):  
H. J. Helgert
Keyword(s):  
Markov Chain ◽  
Transition Probabilities ◽  
Random Variables ◽  
Homogeneous Markov Chain ◽  
Sums Of Random Variables ◽  
Image Position

Assume the sequence of random variables x 0, x 1, x 2, ··· forms a two-state, homogeneous Markov chain with transition probabilities and initial probabilities

scholarly journals A limit theorem for Markov chains with continuous state space

1963 ◽  
Vol 3 (3) ◽  
pp. 351-358 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
State Space ◽  
Transition Probabilities ◽  
Borel Subset ◽  
Homogeneous Markov Chain ◽  
Continuous State Space ◽  
Continuous State ◽  
One Step ◽  
Step Transition ◽  
Image Position ◽  
The Right

Let R denote the set of real numbers, B the σ-field of all Borel subsets of R. A homogeneous Markov Chain with state space a Borel subset Ω of R is a sequence {an}, n≧ 0, of random variables, taking values in Ω, with one-step transition probabilities P(1) (ξ, A) defined by for each choice of ξ, ξ0, …, ξn−1 in ω and all Borel subsets A of ω The fact that the right-hand side of (1.1) does not depend on the ξi, 0 ≧ i > n, is of course the Markovian property, the non-dependence on n is the homogeneity of the chain.

Kolmogorov's differential equations for non-stationary, countable state Markov processes with uniformly continuous transition probabilities

1973 ◽  
Vol 73 (1) ◽  
pp. 119-138 ◽  
Author(s):  
Gerald S. Goodman ◽  
S. Johansen
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
State Space ◽  
Markov Processes ◽  
Transition Probabilities ◽  
The State ◽  
Continuous Transition ◽  
Countable State Space ◽  
Countable State ◽  
Uniformly Continuous

1. SummaryWe shall consider a non-stationary Markov chain on a countable state space E. The transition probabilities {P(s, t), 0 ≤ s ≤ t <t0 ≤ ∞} are assumed to be continuous in (s, t) uniformly in the state i ε E.

Reduction techniques for discrete-time Markov chains on totally ordered state space using stochastic comparisons

2000 ◽  
Vol 37 (03) ◽  
pp. 795-806 ◽  
Author(s):  
Laurent Truffet
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
Stochastic Comparison ◽  
Homogeneous Markov Chain ◽  
Monotone Functions ◽  
Stochastic Comparisons ◽  
Reduction Techniques ◽  
Ordered State

We propose in this paper two methods to compute Markovian bounds for monotone functions of a discrete time homogeneous Markov chain evolving in a totally ordered state space. The main interest of such methods is to propose algorithms to simplify analysis of transient characteristics such as the output process of a queue, or sojourn time in a subset of states. Construction of bounds are based on two kinds of results: well-known results on stochastic comparison between Markov chains with the same state space; and the fact that in some cases a function of Markov chain is again a homogeneous Markov chain but with smaller state space. Indeed, computation of bounds uses knowledge on the whole initial model. However, only part of this data is necessary at each step of the algorithms.

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.

scholarly journals A Diffusion Limit for Generalized Correlated Random Walks

2006 ◽  
Vol 43 (01) ◽  
pp. 60-73 ◽  
Author(s):  
Urs Gruber ◽  
Martin Schweizer
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Partial Sums ◽  
Diffusion Limit ◽  
Correlated Random Walk ◽  
Binomial Trees ◽  
Correlated Random Walks ◽  
Limiting Behaviour ◽  
Image Position

A generalized correlated random walk is a process of partial sums such that (X, Y) forms a Markov chain. For a sequence (X n ) of such processes in which each takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Y n . Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.

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