Convergence of Probability Measures

Wiley Series in Probability and Statistics

by Patrick Billingsley

A new look at weak-convergence methods in metric spaces-from a master of probability theory In this new edition, Patrick Billingsley updates his classic work Convergence of Probability Measures to reflect developments of the past thirty years. Widely known for his straightforward approach and reader-friendly style, Dr. Billingsley presents a clear, precise, up-to-date account of probability limit theory in metric spaces. He incorporates many examples and applications that illustrate the power and utility of this theory in a range of disciplines-from analysis and number theory to statistics, engineering, economics, and population biology. With an emphasis on the simplicity of the mathematics and smooth transitions between topics, the Second Edition boasts major revisions of the sections on dependent random variables as well as new sections on relative measure, on lacunary trigonometric series, and on the Poisson-Dirichlet distribution as a description of the long cycles in permutations and the large divisors of integers. Assuming only standard measure-theoretic probability and metric-space topology, Convergence of Probability Measures provides statisticians and mathematicians with basic tools of probability theory as well as a springboard to the "industrial-strength" literature available today.

Genres: Mathematics

296 pages, Hardcover

First published January 1, 1968

Reviews

[Tinwerume · Rating 3 out of 5]

A little lacking in motivation, but overall quite good

Update: I should note that the notation is *annoyingly* inconsistent/difficult. In particular he uses the same symbol (\Rightarrow) for all forms of convergence, which would normally be fine but in a book with 5 different kinds of convergence running around is kind of inexcusable. (It's almost always weak convergence of measures but still)

Update 2: I checked against my physical copy (which is 1st edition, the ebook copy I have is the second condition) and overall it seems better organized and better about addressing the notation issues. (The particular thing that prompted this was that the second edition is missing an index for a limit which is kind of an egregious error. The first edition has it correctly). So try reading the first edition if you can.

Update 3: I've covered the main chapters. There's quite a bit of non-essential content that I'll probably return in the future, but it's not really "core" convergence. Overall I think it's mediocre, *but* it has the saving grace of being much more comprehensive than any other treatment I know of, so I'm comfortable recommending it. Just get the first edition and be ready for some frustration.

[Kevin K. Gillette · Rating 4 out of 5]

This is the third book of Prof. Billingsley that I've read or perused over the past ten years, and I will say this: The man knows his stuff! This is an outstanding blend of analytic mathematics and probability theory, written at a graduate-student level. Prerequisites for this book would be real analysis, topology (at least point-set topology, although algebraic topology wouldn't hurt), stochastic processes (via Karlin or other), and perhaps at least an introduction to functional analysis.

The writing in this volume, although complete, is also rather terse, so if you are someone who likes a bit more meat on the bones, you would do well to supplement this book with (for example) the classic volumes by Breiman (Probability), Karlin & Taylor (A First Course in Stochastic Processes), Royden (Real Analysis), and Hocking & Young (Topology). Readers familiar with the writing of Lars Ahlfors on the theory of functions of a complex variable (Complex Analysis) will recognize the brevity of exposition in Billingsley's work.

I liked Prof. Billingsley's progression of metric spaces, from C[0,1] to D[0,1] to D[0,∞). From a convergence-theorem perspective, this makes sense, going from the 'tidiest' to the 'messiest'.

Highly recommended!

[David · Rating 4 out of 5]

This is a highly technical book, but a wonderful one. The sheer elegance of the theory explained in the book is actually deeply moving. (Kind of like Galois theory)

The book's awesomeness is somehow enhanced by knowing that Patrick Billingsley is a mean clarinet player. He also played the math teacher in the movie "My Bodyguard". How cool is that?

[Nils Lid Hjort · Rating 5 out of 5]

I am looking through certain things just now, in my copy of the 1968 classic. Chaucer 1387 on page 153. Can I do the arcsine things more simply, than on pages 80-83? At any rate the book remains an inspiration, with clear writing about difficult matters.

Billingsley was also an active *actor*, and knew his Shakespeare.

Having (again) pulled Billingsley off my shelf, I'm reminded of the following, from being interviewed by Ingrid Glad and Ørnulf Borgan (published in Norwegian, in Tilfeldig Gang, then in English, in Scandinavian Journal of Statistics):

Ingrid: Det kan vi se for oss! Og eye-opener nummer 3?

Det var at Jan Hoem, Odd Aalens hovedfagsveileder, hadde hørt at jeg kunne ting som ikke stod i bøkene og artiklene, om diverse rundt Aalen-teoriene. Dermed ble jeg invitert til å holde foredrag i Stockholm, der han var da. Da vi snakket litt på forhånd, stilte han kontrollspørsmålet, «har du lest Billingsley (1968)?». «Ja», sa jeg.

https://www.mn.uio.no/math/english/re...

[Peter Frazier · Rating 5 out of 5]

I am this book right now in order to generalize a proof concerning convergence of posterior measures in sequential Bayesian statistical problems. It is very well written, although it does require a deep level of comfort with measure-theoretic probability. All the general theory seems to be in chapter 1, which is only about 70 pages, so it seems like one can get a good foundation by only reading that. The rest of the book is applications: measures on continuous functions (Billingsley does lots of stuff using the convergence of random walks to a Brownian motion to derive probability laws on the path properties of Brownian motion); measures on right-continuous functions (useful if you are dealing with jump processes); etc. A great book, if you like that sort of thing.