The Role of Weak Convergence in Probability Theory

S.R.S.Varadhan

1. Introduction

The concept of limit theorems in probability goes way back. The ﬁrst limit

theorem, the weak law of large numbers was proved by Jacob Bernoulli [1] in the

early eighteenth century. This was quickly followed by De Moivre [3] who proved the

central limit theorem establishing the approximation of the binomial distribution

by a normal distribution. Further work by, Laplace, Gauss, Levy, Khintichine,

Kolmogorov, Gnedenko and others put limit theorems at the center of probability

theory [11].

The connection between random walks and Brownian motion was understood

by Levy and others along with the idea that distributions of quantities like the

maximum etc, based on random walks, converges to the corresponding distributions

derived from Brownian motion. Doob [6] formulated this more precisely in his

paper on ‘Heuristic approach to the Kolmogorov-Smirnov Theorems’. Donsker [5],

in his thesis, established the ﬁrst general theorem to the eﬀect that Doob’s heuristic

proof can in fact be justiﬁed. However his approach was too dependent on ﬁnite

dimensional approximations.

At this point the study of stochastic processes as probability distributions on

function spaces began. Contributions were made by LeCam [13] in the United

States, Kolmogorov [12], Prohorov [15], Skorohod [17] and others in USSR, as well

as Varadarajan [20] in India. Alexandrov in the 1940’s had studied set functions

on topological spaces and now powerful techniques from functional analysis could

be used to study stochastic processes as measures on function spaces. A random

walk or any stochastic process induces a probability distribution on the space of

paths. By interpolation or some such simple device both the approximating and

the limiting distributions can be put on the same space of paths. The question

then reduces to the investigation of the convergence of a sequence µn of probability

measures on a space X of paths to a limit µ. It is clear that the measures µn,

in the case of random walks, look qualitatively diﬀerent from Brownian paths and

hence µn ⊥ µ. It is not going to be true that µn(A) → µ(A) for all measurable sets

A ∈ X.

Functional analysis now provides a useful window. The space X of paths comes

with a topology. A probability measure µ deﬁnes a normalized non-negative linear

1

Contemporary Mathematics

Volume 490, 2009

c 2009 American Mathematical Society

3

S.R.S.Varadhan

1. Introduction

The concept of limit theorems in probability goes way back. The ﬁrst limit

theorem, the weak law of large numbers was proved by Jacob Bernoulli [1] in the

early eighteenth century. This was quickly followed by De Moivre [3] who proved the

central limit theorem establishing the approximation of the binomial distribution

by a normal distribution. Further work by, Laplace, Gauss, Levy, Khintichine,

Kolmogorov, Gnedenko and others put limit theorems at the center of probability

theory [11].

The connection between random walks and Brownian motion was understood

by Levy and others along with the idea that distributions of quantities like the

maximum etc, based on random walks, converges to the corresponding distributions

derived from Brownian motion. Doob [6] formulated this more precisely in his

paper on ‘Heuristic approach to the Kolmogorov-Smirnov Theorems’. Donsker [5],

in his thesis, established the ﬁrst general theorem to the eﬀect that Doob’s heuristic

proof can in fact be justiﬁed. However his approach was too dependent on ﬁnite

dimensional approximations.

At this point the study of stochastic processes as probability distributions on

function spaces began. Contributions were made by LeCam [13] in the United

States, Kolmogorov [12], Prohorov [15], Skorohod [17] and others in USSR, as well

as Varadarajan [20] in India. Alexandrov in the 1940’s had studied set functions

on topological spaces and now powerful techniques from functional analysis could

be used to study stochastic processes as measures on function spaces. A random

walk or any stochastic process induces a probability distribution on the space of

paths. By interpolation or some such simple device both the approximating and

the limiting distributions can be put on the same space of paths. The question

then reduces to the investigation of the convergence of a sequence µn of probability

measures on a space X of paths to a limit µ. It is clear that the measures µn,

in the case of random walks, look qualitatively diﬀerent from Brownian paths and

hence µn ⊥ µ. It is not going to be true that µn(A) → µ(A) for all measurable sets

A ∈ X.

Functional analysis now provides a useful window. The space X of paths comes

with a topology. A probability measure µ deﬁnes a normalized non-negative linear

1

Contemporary Mathematics

Volume 490, 2009

c 2009 American Mathematical Society

3