## 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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Page 19

Theorem 1.2: The generating function of the number of

generation, given that Xo = 1, is F,(s) = FIFa_1(8)] (1.30) where F1(s) = F(s), ..., F,

11(s) = FIF,(s)] = F,[F(s)]; that is, F,(s) is the nth functional iterate of F(s). Also, if Xo

...

Theorem 1.2: The generating function of the number of

**individuals**in the nthgeneration, given that Xo = 1, is F,(s) = FIFa_1(8)] (1.30) where F1(s) = F(s), ..., F,

11(s) = FIF,(s)] = F,[F(s)]; that is, F,(s) is the nth functional iterate of F(s). Also, if Xo

...

Page 48

The component scalar random variables Xin (i = 1, 2, ..., N) represent the number

of

the random variables X, assumed values in the state space 3: which consisted of

...

The component scalar random variables Xin (i = 1, 2, ..., N) represent the number

of

**individuals**of the ith type in the nth generation. In the one-dimensional casethe random variables X, assumed values in the state space 3: which consisted of

...

Page 185

time t = 0 there are n1

to determine the probability that a

interval [0,t). The models considered in this section are due to Pyke." Let the ...

time t = 0 there are n1

**individuals**in R, and n2**individuals**in Ra. The problem isto determine the probability that a

**individuals**will migrate from R, to R, in the timeinterval [0,t). The models considered in this section are due to Pyke." Let the ...

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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