''Convergence of Probability Measures'' is a graduate textbook in the field of mathematical

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

. It was written by

Patrick Billingsley Patrick Paul Billingsley (May 3, 1925 – April 22, 2011) was an American mathematician and stage and screen actor, noted for his books in advanced probability theory and statistics. He was born and raised in Sioux Falls, South Dakota, and gradu ...

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. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries. Readers are expected to already be familiar with both the fundamentals of probability theory and the

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

s. The subject

weak convergence of measures In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharing ...

involves rigorous study of how a continuous time (or space) stochastic process arises as a

scaling limit In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world process ...

of a discrete time (or space) process. A fundamental example,

Donsker's theorem In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem. Let X_1, X_2, X_3, \ldots be ...

, is convergence of rescaled

random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb ...

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

. The mathematical theory, combining probability and

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

, was first developed in the 1950s by Skorokhod and 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 In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named ...

. It covers key theory tools such as Prokhorov's theorem on relative compactness of measures and the Skorokhod space of càdlàg functions. The second edition includes Skorokhod's representation theorem. Though criticized by

Dudley Dudley is a large market town and administrative centre in the county of West Midlands, England, southeast of Wolverhampton and northwest of Birmingham. Historically an exclave of Worcestershire, the town is the administrative centre of the ...

for insufficient generality, a reviewer wrote "the subject matter is of great current interest and the exposition is lucid and elegant." 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 Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains. Scope Much research involving probability is done under the auspices of applied probability. However, while such res ...

such as

queueing theory Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...

and

empirical process In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state. For a process in a discrete state space a population continuous time Markov chain or Markov population model ...

theory in statistics.

References

Mathematics textbooks 1968 non-fiction books 1999 non-fiction books Probability theory {{mathematics-book-stub