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Probability: Theory and Examples Rick Durrett, Duke U.
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Probability: Theory and Examples Rick Durrett, Duke U.
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Probability: Theory and Examples
Rick Durrett, Duke U.
Version 5, September 15, 2018
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Two dimensional Brownian motion
Copyright 2018, All rights reserved.
ii
Preface
Some times the lights are shining on me. Other times I can
barely see.
Lately it occurs to me what a long strange trip its been.
Grateful Dead
In 1989 when the first edition of the book was completed, my sons
David and Greg were 3 and 1, and the cover picture showed the Dow
Jones at 2650. The last twenty-nine years have brought many changes
but the song remains the same. “The title of the book indicates that as we
develop the theory, we will focus our attention on examples. Hoping that
the book would be a usef ul reference for people who apply probability
in their work, we have tried to emphasize the results that are important
for applications, and illustrated their use with roughly 200 examples.
Probability is not a spectator sport, so the book contains almost 450
exercises to challenge the reader and to deepen their understanding.”
The fifth edition has a number of changes:
• The exercises have been moved to the end of the section. The Ex-
amples, Theorems, and Lemmas are now numbered in one sequence
to make it easier to find things.
• There is a new chapter on multidimensional Brownian motion and
its relationship to PDEs. To make this possible a proof of Itˆo’s
formula has been added to Chapter 7.
• The lengthy Brownian motion chapter has been split into two, with
the second focusing on Donsker’s theorem, etc. The material on
the central limit theorem for martingales and stationary sequences
deleted from the fourth edition has been reinstated.
• The four sections of the random walk chapter have been relocated.
Stopping times have been moved to the martingale chapter; recur-
rence of random walks and the arcsine laws to the Markov chain
chapter; renewal theory has been moved to Chapter 2.
• Some of the exercises that were simply proofs left to the reader, have
been put into the text as lemmas. There are a few new exercises
iii
iv
Typos. The fourth edition contains a list of the people who made cor-
rections to the first three editions. With apologies to those whose contri-
butions I lost track of, this time I need to thank: Richard Arratia, Benson
Au, Swee Hong Chan, Conrado Costa, Nate Eldredge, Steve Evans, Ja-
son Farnon, Christina Goldschmidt, Eduardo Hota, Martin Hildebrand,
Shlomo Leventhal, Jan Lieke, Kyle MacDonald, Ron Peled, Jonathan
Peterson, Erfan Salavati, Byron Schmuland, Timo Seppalainen, Antonio
Carlos de Azevedo Sodre, Shouda Wang, and Ruth Williams. I must con-
fess that Christophe Leuridan pointed one out that I have not corrected.
Lemma 3.4.19 incorrectly asserts that the distributions in its statement
have mean 0, but their means do not exist. The conclusion remains valid
since they are differentiable at 0. A sixth edition is extremely unlikely,
but you can email me about typos and I will post them on my web side.
Family update. As the fourth edition was being completed, David
had recently graduated from Ithaca College and Greg was in his last
semester at MIT applying to graduate school in computer science. Now,
eight years later, Greg has graduated from Berkeley, and is an Assistant
Professor in the Computer Science department at U of Texas in Austin.
Greg works in the field of machine learning, specifically natural language
processing. No, I don’t know what that means but it seems to pay well.
David got his degree in journalism. After an extensive job search process
and some free lance work, David has settled into a steady job working for
a company that produces newsletters for athletic directors and trainers.
In the summer of 2010, Susan and I moved to Durham. Since many
people think that the move was about the weather, I will mention that
during our first summer it was 104 degrees (and humid!) three days in a
row. Yes, it almost never snows here, but when it does, three inches of
snow (typically mixed with ice) will shut down the whole town for four
days. It to ok some time for us to adjust to the Durham/Chapel area,
which has about 10 times as many people as Ithaca and is criss-crossed
by freeways, but we live in a nice quiet neighborhood near the campus.
Susan enjoys volunteering at the Sarah P. Duke gardens and listening to
their talks about the plants of North Carolina and future plans for the
gardens.
As I write this it is the last week before school starts.
Rick Durrett, summer 2018
Conte nts
1 Measure Theory 1
1.1 Probability Spaces . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Random Variables . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Properties of the Integral . . . . . . . . . . . . . . . . . . . 24
1.6 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6.1 Inequalities . . . . . . . . . . . . . . . . . . . . . . 29
1.6.2 Integration to the Limit . . . . . . . . . . . . . . . 30
1.6.3 Computing Expected Values . . . . . . . . . . . . . 32
1.7 Product Measures, Fubini’s Theorem . . . . . . . . . . . . 37
2 Laws of Large Numbers 43
2.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1.1 Sufficient Conditions for Independence . . . . . . . 45
2.1.2 Independence, Distribution, and Expectation . . . . 48
2.1.3 Sums of Independent Random Variables . . . . . . 49
2.1.4 Constructing Independent Random Variables . . . . 52
2.2 Weak Laws of Large Numbers . . . . . . . . . . . . . . . . 56
2.2.1 L
2
Weak Laws . . . . . . . . . . . . . . . . . . . . . 56
2.2.2 Triangular Arrays . . . . . . . . . . . . . . . . . . . 59
2.2.3 Truncation . . . . . . . . . . . . . . . . . . . . . . . 62
2.3 Borel-Cantelli Lemmas . . . . . . . . . . . . . . . . . . . . 67
2.4 Strong Law of Large Numbers . . . . . . . . . . . . . . . . 76
2.5 Convergence of Random Series* . . . . . . . . . . . . . . . 81
2.5.1 Rates of Convergence . . . . . . . . . . . . . . . . . 87
2.5.2 Infinite Mean . . . . . . . . . . . . . . . . . . . . . 88
2.6 Renewal Theory* . . . . . . . . . . . . . . . . . . . . . . . 91
2.7 Large Deviations* . . . . . . . . . . . . . . . . . . . . . . . 105
3 Central Limit Theorems 113
3.1 The De Moivre-Laplace Theorem . . . . . . . . . . . . . . 113
3.2 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . 116
3.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . 116
v
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