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Book DetailsContemporary MathematicsVolume: 50; 1986MSC: Primary 60;
These twentysix expository papers on random matrices and products of random matrices survey the major results of the last thirty years. They reflect both theoretical and applied concerns in fields as diverse as computer science, probability theory, mathematical physics, and population biology. Many of the articles are tutorial, consisting of examples, sketches of proofs, and interpretations of results. They address a wide audience of mathematicians and scientists who have an elementary knowledge of probability theory and linear algebra, but not necessarily any prior exposure to this specialized area. More advanced articles, aimed at specialists in allied areas, survey current research with references to the original literature.
The book's major topics include the computation and behavior under perturbation of Lyapunov exponents and the spectral theory of large random matrices. The applications to mathematical and physical sciences under consideration include computer image generation, card shuffling, and other random walks on groups, Markov chains in random environments, the random Schroedinger equations and random waves in random media.
Most of the papers were originally presented at an AMSIMSSIAM Joint Summer Research Conference held at Bowdoin College in June, 1984. Of special note are the papers by Kotani on random Schroedinger equations, Yin and Bai on spectra for large random matrices, and Newman on the relations between the Lyapunov and eigenvalue spectra. 
Table of Contents

I. Basic theory of products of random matrices [ MR 841077 ]

A. Overviews [ MR 841077 ]

Joseph C. Watkins  Limit theorems for products of random matrices: a comparison of two points of view [ MR 841078 ]

Joel E. Cohen, Harry Kesten and Charles M. Newman  Oseledec’s multiplicative ergodic theorem: a proof [ MR 841079 ]

Y. Guivarc’h and A. Raugi  Products of random matrices: convergence theorems [ MR 841080 ]

F. Ledrappier  Examples of applications of Oseledec’s theorem [ MR 841081 ]

B. Perturbation theory [ MR 841077 ]

Yuri Kifer  Multiplicative ergodic theorems for random diffeomorphisms [ MR 841082 ]

Steve Pincus  FurstenbergKesten results: asymptotic analysis [ MR 841083 ]

Eric V. Slud  Stability of exponential growth rate for randomly perturbed random matrix products via Markovchain arguments [ MR 841084 ]

Volker Wihstutz  Representation, positivity and expansion of Lyapunov exponents for linear stochastic systems [ MR 841085 ]

C. Theory of matrix products [ MR 841077 ]

Maciej Wojtkowski  On uniform contraction generated by positive matrices [ MR 841086 ]

D. Connections with spectral theory [ MR 841077 ]

C. M. Newman  Lyapunov exponents for some products of random matrices: exact expressions and asymptotic distributions [ MR 841087 ]

II. Spectral theory of random matrices [ MR 841077 ]

ChiiRuey Hwang  A brief survey on the spectral radius and the spectral distribution of large random matrices with i.i.d. entries [ MR 841088 ]

Jack W. Silverstein  Eigenvalues and eigenvectors of largedimensional sample covariance matrices [ MR 841089 ]

Y. Q. Yin and Z. D. Bai  Spectra for largedimensional random matrices [ MR 841090 ]

III. Applications to computer science, probability and statistics of products of random matrices [ MR 841077 ]

A. Applications to computer science and statistics [ MR 841077 ]

Persi Diaconis and Mehrdad Shahshahani  Products of random matrices and computer image generation [ MR 841091 ]

Persi Diaconis and Mehrdad Shahshahani  Products of random matrices as they arise in the study of random walks on groups [ MR 841092 ]

B. Applications to Markov chains in random environments [ MR 841077 ]

Robert Cogburn  On products of random stochastic matrices [ MR 841093 ]

M. Rosenblatt  Convolution sequences of measures on the semigroup of stochastic matrices [ MR 841094 ]

Tze Chien Sun  Random walks on semigroups [ MR 841095 ]

Other appliations to probability theory [ MR 841077 ]

Thomas Kaijser  A note on random systems with complete connections and their applications to products of random matrices [ MR 841096 ]

Eric S. Key  Using random matrices to give recurrence and transience criteria for random walk in a random environment [ MR 841097 ]

Gérard Letac  A contraction principle for certain Markov chains and its applications [ MR 841098 ]

IV. Scientific applications of random matrices and their products [ MR 841077 ]

S. Kotani  Lyapunov exponents and spectra for onedimensional random Schrödinger operators [ MR 841099 ]

Robert S. Maier  The density of states of random Schroedinger operators [ MR 841100 ]

M. L. Mehta  Random matrices in nuclear physics and number theory [ MR 841101 ]

George C. Papanicolaou  Random matrices and waves in random media [ MR 841102 ]

Shripad Tuljapurkar  Demographic applications of random matrix products [ MR 841103 ]

Supplements [ MR 841077 ]

Joel E. Cohen, Harry Kesten and Charles M. Newman  Open problems [ MR 841104 ]

Joel E. Cohen  Products of random matrices and related topics in mathematics and science: a bibliography [ MR 841105 ]


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 Table of Contents

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These twentysix expository papers on random matrices and products of random matrices survey the major results of the last thirty years. They reflect both theoretical and applied concerns in fields as diverse as computer science, probability theory, mathematical physics, and population biology. Many of the articles are tutorial, consisting of examples, sketches of proofs, and interpretations of results. They address a wide audience of mathematicians and scientists who have an elementary knowledge of probability theory and linear algebra, but not necessarily any prior exposure to this specialized area. More advanced articles, aimed at specialists in allied areas, survey current research with references to the original literature.
The book's major topics include the computation and behavior under perturbation of Lyapunov exponents and the spectral theory of large random matrices. The applications to mathematical and physical sciences under consideration include computer image generation, card shuffling, and other random walks on groups, Markov chains in random environments, the random Schroedinger equations and random waves in random media.
Most of the papers were originally presented at an AMSIMSSIAM Joint Summer Research Conference held at Bowdoin College in June, 1984. Of special note are the papers by Kotani on random Schroedinger equations, Yin and Bai on spectra for large random matrices, and Newman on the relations between the Lyapunov and eigenvalue spectra.

I. Basic theory of products of random matrices [ MR 841077 ]

A. Overviews [ MR 841077 ]

Joseph C. Watkins  Limit theorems for products of random matrices: a comparison of two points of view [ MR 841078 ]

Joel E. Cohen, Harry Kesten and Charles M. Newman  Oseledec’s multiplicative ergodic theorem: a proof [ MR 841079 ]

Y. Guivarc’h and A. Raugi  Products of random matrices: convergence theorems [ MR 841080 ]

F. Ledrappier  Examples of applications of Oseledec’s theorem [ MR 841081 ]

B. Perturbation theory [ MR 841077 ]

Yuri Kifer  Multiplicative ergodic theorems for random diffeomorphisms [ MR 841082 ]

Steve Pincus  FurstenbergKesten results: asymptotic analysis [ MR 841083 ]

Eric V. Slud  Stability of exponential growth rate for randomly perturbed random matrix products via Markovchain arguments [ MR 841084 ]

Volker Wihstutz  Representation, positivity and expansion of Lyapunov exponents for linear stochastic systems [ MR 841085 ]

C. Theory of matrix products [ MR 841077 ]

Maciej Wojtkowski  On uniform contraction generated by positive matrices [ MR 841086 ]

D. Connections with spectral theory [ MR 841077 ]

C. M. Newman  Lyapunov exponents for some products of random matrices: exact expressions and asymptotic distributions [ MR 841087 ]

II. Spectral theory of random matrices [ MR 841077 ]

ChiiRuey Hwang  A brief survey on the spectral radius and the spectral distribution of large random matrices with i.i.d. entries [ MR 841088 ]

Jack W. Silverstein  Eigenvalues and eigenvectors of largedimensional sample covariance matrices [ MR 841089 ]

Y. Q. Yin and Z. D. Bai  Spectra for largedimensional random matrices [ MR 841090 ]

III. Applications to computer science, probability and statistics of products of random matrices [ MR 841077 ]

A. Applications to computer science and statistics [ MR 841077 ]

Persi Diaconis and Mehrdad Shahshahani  Products of random matrices and computer image generation [ MR 841091 ]

Persi Diaconis and Mehrdad Shahshahani  Products of random matrices as they arise in the study of random walks on groups [ MR 841092 ]

B. Applications to Markov chains in random environments [ MR 841077 ]

Robert Cogburn  On products of random stochastic matrices [ MR 841093 ]

M. Rosenblatt  Convolution sequences of measures on the semigroup of stochastic matrices [ MR 841094 ]

Tze Chien Sun  Random walks on semigroups [ MR 841095 ]

Other appliations to probability theory [ MR 841077 ]

Thomas Kaijser  A note on random systems with complete connections and their applications to products of random matrices [ MR 841096 ]

Eric S. Key  Using random matrices to give recurrence and transience criteria for random walk in a random environment [ MR 841097 ]

Gérard Letac  A contraction principle for certain Markov chains and its applications [ MR 841098 ]

IV. Scientific applications of random matrices and their products [ MR 841077 ]

S. Kotani  Lyapunov exponents and spectra for onedimensional random Schrödinger operators [ MR 841099 ]

Robert S. Maier  The density of states of random Schroedinger operators [ MR 841100 ]

M. L. Mehta  Random matrices in nuclear physics and number theory [ MR 841101 ]

George C. Papanicolaou  Random matrices and waves in random media [ MR 841102 ]

Shripad Tuljapurkar  Demographic applications of random matrix products [ MR 841103 ]

Supplements [ MR 841077 ]

Joel E. Cohen, Harry Kesten and Charles M. Newman  Open problems [ MR 841104 ]

Joel E. Cohen  Products of random matrices and related topics in mathematics and science: a bibliography [ MR 841105 ]