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Papers by David Williams

Research paper thumbnail of Book Review: Diffusions, Markov processes, and martingales, Volume One: Foundations, {Second Edition}

Bulletin of the American Mathematical Society, 1997

Research paper thumbnail of The process extended to the boundary

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1964

Research paper thumbnail of Brownian motions of ellipsoids

Transactions of the American Mathematical Society, 1986

The object of this paper is to provide an elementary treatment (involving no differential geometr... more The object of this paper is to provide an elementary treatment (involving no differential geometry) of Brownian motions of ellipsoids, and, in particular, of some remarkable results first obtained by Dynkin. The canonical right-invariant Brownian motion G = { G ( t ) } G = \{ G(t)\} on GL ( n ) {\text {GL}}(n) induces processes X = G G T X = G{G^T} and Y = G T G Y = {G^T}G on the space of positive-definite symmetric matrices. The motion of the common eigenvalues of X X and Y Y is analysed. It is further shown that the orthonormal frame of eigenvectors of X X ultimately behaves like Brownian motion on O ( n ) {\text {O}}(n) , while that of Y Y converges to a limiting value. The Y Y process is that studied by Dynkin and Orihara. From a naive standpoint, the X X process would seem to provide a more natural model.

Research paper thumbnail of Weighing the Odds: A Course in Probability and Statistics

The American Mathematical Monthly, 2003

In this lively look at both subjects, David Williams convinces Mathematics students of the intrin... more In this lively look at both subjects, David Williams convinces Mathematics students of the intrinsic interest of Statistics and Probability, and Statistics students that the language of Mathematics can bring real insight and clarity to their subject. He helps students build the ...

Research paper thumbnail of Book Review: Multidimensional diffusion processes

Bulletin of the American Mathematical Society, 1980

BOOK REVIEWS equicoloration theorem has been shown by Hajnal and Szemeredi. There are many other ... more BOOK REVIEWS equicoloration theorem has been shown by Hajnal and Szemeredi. There are many other subjects covered in this volume and we shall not attempt to enumerate them here. The topics covered are generally discussed in depth. The book, though self-contained, would be difficult reading without some prior basic knowledge of Graph Theory. The pace is brisk and the reader is quickly brought to the frontiers of the subject. Bollobâs is a fastidious writer. The theorems are precisely stated and the proofs are carefully written. The publisher, Academic Press, has done a fine job. Most important, Bollobâs is a mathematician who knows his material. In section after section he takes a set of theorems and, by appropriate concatenation plus some well chosen words of explanation, he creates a Theory.

Research paper thumbnail of Wiener-hopf factorization for matrices

Lecture Notes in Mathematics, 1980

Research paper thumbnail of Path Decomposition and Continuity of Local Time for One-Dimensional Diffusions, I

Proceedings of the London Mathematical Society, 1974

1.2. Trotter's theorem. We use the following notation. 3& is the Borel a-algebra &(R) on R. C is ... more 1.2. Trotter's theorem. We use the following notation. 3& is the Borel a-algebra &(R) on R. C is the space of all (bounded and unbounded) continuous functions from [0, oo) to R. The symbol w will be used to denote a typical element of G.

Research paper thumbnail of On the construction problem for Markov chains

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1964

Research paper thumbnail of Fictitious states, coupled laws and local time

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1969

Research paper thumbnail of A simple PDE and Wiener-Hopf Riccati equations

Communications on Pure and Applied Mathematics, 2005

Research paper thumbnail of Book Review: Diffusions, Markov processes, and martingales, Volume One: Foundations, {Second Edition}

Bulletin of the American Mathematical Society, 1997

Research paper thumbnail of The process extended to the boundary

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1964

Research paper thumbnail of Brownian motions of ellipsoids

Transactions of the American Mathematical Society, 1986

The object of this paper is to provide an elementary treatment (involving no differential geometr... more The object of this paper is to provide an elementary treatment (involving no differential geometry) of Brownian motions of ellipsoids, and, in particular, of some remarkable results first obtained by Dynkin. The canonical right-invariant Brownian motion G = { G ( t ) } G = \{ G(t)\} on GL ( n ) {\text {GL}}(n) induces processes X = G G T X = G{G^T} and Y = G T G Y = {G^T}G on the space of positive-definite symmetric matrices. The motion of the common eigenvalues of X X and Y Y is analysed. It is further shown that the orthonormal frame of eigenvectors of X X ultimately behaves like Brownian motion on O ( n ) {\text {O}}(n) , while that of Y Y converges to a limiting value. The Y Y process is that studied by Dynkin and Orihara. From a naive standpoint, the X X process would seem to provide a more natural model.

Research paper thumbnail of Weighing the Odds: A Course in Probability and Statistics

The American Mathematical Monthly, 2003

In this lively look at both subjects, David Williams convinces Mathematics students of the intrin... more In this lively look at both subjects, David Williams convinces Mathematics students of the intrinsic interest of Statistics and Probability, and Statistics students that the language of Mathematics can bring real insight and clarity to their subject. He helps students build the ...

Research paper thumbnail of Book Review: Multidimensional diffusion processes

Bulletin of the American Mathematical Society, 1980

BOOK REVIEWS equicoloration theorem has been shown by Hajnal and Szemeredi. There are many other ... more BOOK REVIEWS equicoloration theorem has been shown by Hajnal and Szemeredi. There are many other subjects covered in this volume and we shall not attempt to enumerate them here. The topics covered are generally discussed in depth. The book, though self-contained, would be difficult reading without some prior basic knowledge of Graph Theory. The pace is brisk and the reader is quickly brought to the frontiers of the subject. Bollobâs is a fastidious writer. The theorems are precisely stated and the proofs are carefully written. The publisher, Academic Press, has done a fine job. Most important, Bollobâs is a mathematician who knows his material. In section after section he takes a set of theorems and, by appropriate concatenation plus some well chosen words of explanation, he creates a Theory.

Research paper thumbnail of Wiener-hopf factorization for matrices

Lecture Notes in Mathematics, 1980

Research paper thumbnail of Path Decomposition and Continuity of Local Time for One-Dimensional Diffusions, I

Proceedings of the London Mathematical Society, 1974

1.2. Trotter's theorem. We use the following notation. 3& is the Borel a-algebra &(R) on R. C is ... more 1.2. Trotter's theorem. We use the following notation. 3& is the Borel a-algebra &(R) on R. C is the space of all (bounded and unbounded) continuous functions from [0, oo) to R. The symbol w will be used to denote a typical element of G.

Research paper thumbnail of On the construction problem for Markov chains

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1964

Research paper thumbnail of Fictitious states, coupled laws and local time

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1969

Research paper thumbnail of A simple PDE and Wiener-Hopf Riccati equations

Communications on Pure and Applied Mathematics, 2005