Homogeneous Denumerable Markov Processes
Markov processes play an important role in the study of probability theory. Homogeneous denumerable Markov processes are among the main topics in the theory and have a wide range of application in various fields of science and technology (for example, in physics, cybernetics, queuing theory and dynamical programming). This book is a detailed presentation and summary of the research results obtained by the authors in recent years. Most of the results are published for the first time. Two new methods are given: one is the minimal nonnegative solution, the second the limit transition method. With the help of these two methods, the authors solve many important problems in the framework of denumerable Markov processes.
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The Second Construction Theorem
Theory of Minimal Nonnegative Solutions for Systems
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1-bounded equations a(co ae(dX)e atomic almost closed boundary theory called Cauchy sequence Chapter cikxk closed set coeQ Corollary deduce Definition denote density matrix Denumerable Markov Processes determined uniquely distribution and moments dX)e equations xt excessive function exist infinitely F-type Q-process first-type system flying point harmonic function harmonic measure Hence homogeneous denumerable Markov honest i,jeE ieE jeE Lemma linearly independent Markov chain Markov property matrix of order minimal nonnegative solution minimal process necessary and sufficient nonnegative linear equations normal system O-type obtain pij(t probability space processes of order proof of Theorem pseudo-normal system Q is conservative Q is nonconservative Q-matrix is given Q-process is unique random variable second-type strictly nonhomogeneous equations sufficient condition Suppose a Q-matrix system of 1-bounded system of equations system of homogeneous system of nonnegative system of strictly transition probability matrix unique ordinary X(co zero solution