View source for Convergence of Probability Measures

{{Short description|Textbook by Patrick Billingsley}}{{Infobox book
| author            = [[Patrick Billingsley]]
| pub_date          = 1968
| publisher         = Wiley
| genre             = Textbook
}}

{{italic title}}
'''''Convergence of Probability Measures''''' 
is a graduate textbook in the field of mathematical [[probability theory]]. 
It was written by [[Patrick Billingsley]] and published by Wiley in 1968. A second edition in 1999 both simplified its treatment of previous topics and updated the book for more recent developments.{{r|schweizer}} The Basic Library List Committee of the [[Mathematical Association of America]] has recommended its inclusion in undergraduate mathematics libraries.{{r|bll}} Readers are expected to already be familiar with both the fundamentals of probability theory and the [[topology]] of [[metric space]]s.{{r|theodorescu}}

The subject [[Convergence of measures|weak convergence of measures]] involves 
rigorous study of how a continuous time (or space) stochastic process arises as a
[[scaling limit]] of a discrete time (or space)  process.
A fundamental example, [[Donsker's theorem]], is convergence of 
rescaled [[random walk]] to [[Wiener process|Brownian motion]]. 
The mathematical theory, combining probability and [[functional analysis]],
was first developed in the 1950s by  [[Anatoliy Skorokhod|Skorokhod]] and [[Yuri Prokhorov|Prokhorov]], but was regarded as a specialized advanced topic. This book's contribution was a self-contained treatment at a useful basic level of abstraction, that of [[Polish space]]. It covers key theory tools such as [[Prokhorov's theorem]] on relative compactness of measures and the [[Càdlàg|Skorokhod space of càdlàg functions]]. The second edition includes
[[Skorokhod's representation theorem]].  Though criticized by [[Richard M. Dudley|Dudley]] for insufficient generality,{{r|dudley}} a reviewer wrote "the subject matter is of great current interest and the exposition is lucid and elegant."{{r|MR}} By being widely accessible it was for many years the standard reference, as evidenced by over 22,000 citations on Google Scholar.  In particular, the subject became a highly valuable tool within burgeoning fields of [[applied probability]] such as [[queueing theory]]{{r|iglehart}} and [[empirical process]] theory in statistics.{{r|shorack}}

==References==
{{reflist|refs=
 
<ref name=bll>{{citation|url=https://www.maa.org/press/maa-reviews/convergence-of-probability-measures|title=Convergence of Probability Measures|type=listing but no review|work=MAA Reviews|publisher=Mathematical Association of America|access-date=2021-01-24}}</ref>

<ref name=dudley>{{citation|last=Dudley|first=Richard M.|journal=Bulletin of the American Mathematical Society|title=Review of ''Convergence of probability measures'' (1st ed.)|volume=77|pages=25–27|year=1971|doi=10.1090/S0002-9904-1971-12602-2|doi-access=free}}</ref>

<ref name=iglehart>{{citation|last=Iglehart|first=D.L.|title= Weak convergence in queueing theory|journal=Advances in Applied Probability|volume=5|pages=570–594|year=1973|doi=10.2307/1425835|jstor=1425835}}</ref>

<ref name=schweizer>{{citation|last=Schweizer|first=M.|title=Review of ''Convergence of Probability Measures'' (2nd ed.)|work=[[zbMATH]]|zbl=0944.60003}}</ref>

<ref name=shorack>{{citation |author1-link=Galen Shorack |author2-link=Jon A. Wellner |last1=Shorack |first=Galen R. |last2=Wellner |first2=Jon A. |title=Empirical Processes with Applications to Statistics |publisher=Wiley |year=1986 |isbn=0-471-86725-X }}</ref>

<ref name=theodorescu>{{citation|last=Theodorescu|first=R.|title=Review of ''Convergence of Probability Measures'' (1st ed.)|work=[[zbMATH]]|zbl=0172.21201}}</ref>

<ref name=MR>{{citation
 | last = Siddiqui | first = M.M.
 | journal = [[MathSciNet]]
 | mr = 0233396 
 | title = none
 | year = 1969}}</ref>

}}

[[Category:Mathematics textbooks]]
[[Category:1968 non-fiction books]]
[[Category:1999 non-fiction books]]
[[Category:Probability theory]]


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