# Probability Approximations via the Poisson Clumping Heuristic

• David Aldous
Book

Part of the Applied Mathematical Sciences book series (AMS, volume 77)

1. Front Matter
Pages i-xv
2. David Aldous
Pages 1-22
3. David Aldous
Pages 23-43
4. David Aldous
Pages 44-71
5. David Aldous
Pages 72-105
6. David Aldous
Pages 106-117
7. David Aldous
Pages 118-130
8. David Aldous
Pages 131-148
9. David Aldous
Pages 149-166
10. David Aldous
Pages 167-189
11. David Aldous
Pages 190-219
12. David Aldous
Pages 220-236
13. David Aldous
Pages 237-245
14. David Aldous
Pages 246-251
15. David Aldous
Pages 252-252
16. Back Matter
Pages 253-271

### Introduction

If you place a large number of points randomly in the unit square, what is the distribution of the radius of the largest circle containing no points? Of the smallest circle containing 4 points? Why do Brownian sample paths have local maxima but not points of increase, and how nearly do they have points of increase? Given two long strings of letters drawn i. i. d. from a finite alphabet, how long is the longest consecutive (resp. non-consecutive) substring appearing in both strings? If an imaginary particle performs a simple random walk on the vertices of a high-dimensional cube, how long does it take to visit every vertex? If a particle moves under the influence of a potential field and random perturbations of velocity, how long does it take to escape from a deep potential well? If cars on a freeway move with constant speed (random from car to car), what is the longest stretch of empty road you will see during a long journey? If you take a large i. i. d. sample from a 2-dimensional rotationally-invariant distribution, what is the maximum over all half-spaces of the deviation between the empirical and true distributions? These questions cover a wide cross-section of theoretical and applied probability. The common theme is that they all deal with maxima or min­ ima, in some sense.

### Keywords

Brownian motion Markov chain Maxima hitting time random walk

#### Authors and affiliations

• David Aldous
• 1
1. 1.Department of StatisticsUniversity of California-BerkeleyBerkeleyUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4757-6283-9
• Copyright Information Springer-Verlag New York 1989
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4419-3088-0
• Online ISBN 978-1-4757-6283-9
• Series Print ISSN 0066-5452
• Buy this book on publisher's site
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