Probability Approximations via the Poisson Clumping Heuristic

1. Front Matter

   Pages i-xv

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2. The Heuristic

   • David Aldous

   Pages 1-22

3. Markov Chain Hitting Times

   • David Aldous

   Pages 23-43

4. Extremes of Stationary Processes

   • David Aldous

   Pages 44-71

5. Extremes of Locally Brownian Processes

   • David Aldous

   Pages 72-105

6. Simple Combinatorics

   • David Aldous

   Pages 106-117

7. Combinatorics for Processes

   • David Aldous

   Pages 118-130

8. Exponential Combinatorial Extrema

   • David Aldous

   Pages 131-148

9. Stochastic Geometry

   • David Aldous

   Pages 149-166

10. Multi-Dimensional Diffusions

    • David Aldous

    Pages 167-189

11. Random Fields

    • David Aldous

    Pages 190-219

12. Brownian Motion: Local Distributions

    • David Aldous

    Pages 220-236

13. Miscellaneous Examples

    • David Aldous

    Pages 237-245

14. The Eigenvalue Method

    • David Aldous

    Pages 246-251

15. Postscript

    • David Aldous

    Pages 252-252

16. Back Matter

    Pages 253-271

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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.

• Brownian motion
• Markov chain
• Maxima
• hitting time
• random walk

• Department of Statistics, University of California-Berkeley, Berkeley, USA

  David Aldous