## Elements of the Theory of Markov Processes and Their ApplicationsGraduate-level text and reference in probability, with numerous scientific applications. Nonmeasure-theoretic introduction to theory of Markov processes and to mathematical models based on the theory. Appendixes. Bibliographies. 1960 edition. |

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Results 1-3 of 53

Page 14

Definition: A state i is

state j (#i) such that p} > 0 implies the existence of a positive integer m such that p

;"> 0. A state which is not essential is

Definition: A state i is

**called**essential if the existence of a positive integer n and astate j (#i) such that p} > 0 implies the existence of a positive integer m such that p

;"> 0. A state which is not essential is

**called**inessential. Definition: Let i and j ...Page 17

Given that X, − i, we introduce a random variable T, defined as follows: T. = m if X,

is # i for 1 s k < m and Xar, H i. From (1.13) we see that < 9(T. = n} = K? The

random variable T, is

can ...

Given that X, − i, we introduce a random variable T, defined as follows: T. = m if X,

is # i for 1 s k < m and Xar, H i. From (1.13) we see that < 9(T. = n} = K? The

random variable T, is

**called**the recurrence time for the state i. It is clear that T.can ...

Page 143

In these cases no boundary conditions have to be imposed, and the boundaries

are

probability of taking the value r, or ra. For these processes there are two possible

...

In these cases no boundary conditions have to be imposed, and the boundaries

are

**called**natural. Second, the coefficients may be such that X(t) has positiveprobability of taking the value r, or ra. For these processes there are two possible

...

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### Contents

Introduction | 1 |

Processes Discrete in Space and Continuous in Time | 57 |

Processes Continuous in Space and Time | 129 |

Copyright | |

9 other sections not shown

### Other editions - View all

Elements of the Theory of Markov Processes and Their Applications A. T. Bharucha-Reid Limited preview - 2012 |

Elements of the Theory of Markov Processes and Their Applications Albert T. Bharucha-Reid Limited preview - 1997 |

### Common terms and phrases

absorber addition applications approach arrival associated assume assumptions becomes birth boundary branching processes called cascade coefficients collision concerned condition consider constant continuous counter death defined denote density derive described determined developed differential equation diffusion discussion distribution function electron energy epidemic equal exists expected expression finite fluctuation given gives growth Hence independent individuals initial condition integral interest interval introduce Kolmogorov equations Laplace transform length limit machine Markov Markov chain Markov processes Math mathematical mean method moments necessary nucleon obtain particle particular photon Poisson population positive primary problem Proof properties queueing radiation random variable reaction refer relation represent respectively satisfies shown simple ſº solution Statist Stochastic Processes Theorem theory tion transition probabilities zero