The Role of Weak Convergence in Probability Theory
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
Contemporary Mathematics
Volume 490, 2009
c 2009 American Mathematical Society
Previous Page Next Page