Non-negative Matrices and Markov ChainsSpringer Science & Business Media, 2 lug 2006 - 281 pagine Since its inception by Perron and Frobenius, the theory of non-negative matrices has developed enormously and is now being used and extended in applied fields of study as diverse as probability theory, numerical analysis, demography, mathematical economics, and dynamic programming, while its development is still proceeding rapidly as a branch of pure mathematics in its own right. While there are books which cover this or that aspect of the theory, it is nevertheless not uncommon for workers in one or another branch of its development to be unaware of what is known in other branches, even though there is often formal overlap. One of the purposes of this book is to relate several aspects of the theory, insofar as this is possible. The author hopes that the book will be useful to mathematicians; but in particular to the workers in applied fields, so the mathematics has been kept as simple as could be managed. The mathematical requisites for reading it are: some knowledge of real-variable theory, and matrix theory; and a little knowledge of complex-variable; the emphasis is on real-variable methods. (There is only one part of the book, the second part of 55.5, which is of rather specialist interest, and requires deeper knowledge.) Appendices provide brief expositions of those areas of mathematics needed which may be less g- erally known to the average reader. |
Sommario
| 1 | |
CHAPTER 2 | 29 |
CHAPTER 3 | 80 |
Markov Chains and Finite Stochastic Matrices | 112 |
CHAPTER 5 | 161 |
CHAPTER 6 | 199 |
CHAPTER 7 | 221 |
Appendix A Some Elementary Number Theory | 247 |
| 271 | |
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A₁ a₁j analogous aperiodic asymptotic Bibliography and Discussion canonical form Chapter coefficient of ergodicity condition convergence Corollary corresponding countable defined definition diagonal elements elementwise equation Exercise exists Fatou's Lemma fixed follows forward products Frobenius G₁ greatest common divisor H₁ Hence incidence matrix index set inequality inessential integers invariant measure irreducible irreducible matrix Lemma Linear Algebra Markov chains Math metric space n x n non-negative matrices null-recurrent P₁ Perron Perron-Frobenius eigenvalue Perron-Frobenius Theorem positive integer positive-recurrent primitive primitive matrix probabilistic probability vector proof of Theorem R-positive r₁ recurrent result row sums Sarymsakov satisfying scrambling Seneta sequence Show single essential class stationary distribution stationary distribution vector stochastic matrices strong ergodicity subinvariant measure substochastic superregular Suppose T₁ Theorem 1.1 theory transient transition matrix truncation unique stationary distribution v₁ weak ergodicity x₁ zero