*James Oxley*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0015
- Subject:
- Mathematics, Probability / Statistics

Dominic Welsh began writing papers in matroid theory nearly forty years ago. Since then, he has made numerous important contributions to the subject. This chapter reviews Dominic Welsh's work in and ...
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Dominic Welsh began writing papers in matroid theory nearly forty years ago. Since then, he has made numerous important contributions to the subject. This chapter reviews Dominic Welsh's work in and influence on the development of matroid theory.Less

Dominic Welsh began writing papers in matroid theory nearly forty years ago. Since then, he has made numerous important contributions to the subject. This chapter reviews Dominic Welsh's work in and influence on the development of matroid theory.

*Geoffrey Grimmett and Colin McDiarmid (eds)*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.001.0001
- Subject:
- Mathematics, Probability / Statistics

Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and ...
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Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and inspired generations of students and researchers in mathematics. This book summarizes and reviews the consistent themes from his work through a series of articles written by renowned experts. These articles, presented as chapters, contain original research work, set in a broader context by the inclusion of review material.Less

Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and inspired generations of students and researchers in mathematics. This book summarizes and reviews the consistent themes from his work through a series of articles written by renowned experts. These articles, presented as chapters, contain original research work, set in a broader context by the inclusion of review material.

*David Stirzaker*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0018
- Subject:
- Mathematics, Probability / Statistics

This chapter examines a random process (X(t):t ≥ 0) taking values in R, that is governed by the events of an independent renewal process N(t), as follows: whenever an event of N(t) occurs, the ...
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This chapter examines a random process (X(t):t ≥ 0) taking values in R, that is governed by the events of an independent renewal process N(t), as follows: whenever an event of N(t) occurs, the process X(t) is restarted and runs independently of the past with initial value that has the same distribution as X(0). The case when each segment of the process between consecutive events of N(t) is a diffusion is studied, and expressions for the characteristic function of X(t) and its stationary distribution as t → ∞ are presented. An expression is derived for the expected first-passage time of X(t) to any value a, and several explicit examples of interest are considered. The chapter presents two approaches: first, it uses Wald's equation which supplies the mean in quite general circumstances; second, it explores possibilities for use of the moment-generating function of the first-passage time.Less

This chapter examines a random process (*X*(*t*):*t* ≥ 0) taking values in R, that is governed by the events of an independent renewal process *N*(*t*), as follows: whenever an event of *N*(*t*) occurs, the process *X*(*t*) is restarted and runs independently of the past with initial value that has the same distribution as *X*(0). The case when each segment of the process between consecutive events of *N*(*t*) is a diffusion is studied, and expressions for the characteristic function of *X*(*t*) and its stationary distribution as t → ∞ are presented. An expression is derived for the expected first-passage time of *X*(*t*) to any value *a*, and several explicit examples of interest are considered. The chapter presents two approaches: first, it uses Wald's equation which supplies the mean in quite general circumstances; second, it explores possibilities for use of the moment-generating function of the first-passage time.