2 - Stochastic Convergence

Summary

This chapter provides a review of basic modes of convergence of sequences of stochastic vectors, in particular convergence in distribution and in probability.

Basic Theory

A random vector in is a vector X = (X₁, … , X_k) of real random variables. t The distributionfunction of X is the map

A sequence of random vectors Xn is said to converge in distribution to a random vector X if

for every x at which the limit distribution function is continuous. Alternative names are weak convergence and convergence in law. As the last name suggests, the convergence only depends on the induced laws of the vectors and not on the probability spaces on which they are defined. Weak convergence is denoted by if X has distribution L, or a distribution with a standard code, such as N(O, 1), then also by

Let d (x, y) be a distance function on IRk that generates the usual topology. For instance, the Euclidean distance

A sequence of random variables Xn is said to converge in probability to X if for all

This is denoted by. In this notation convergence in probability is the same as

As we shall see, convergence in probability is stronger than convergence in distribution. An even stronger mode of convergence is almost-sure convergence. The sequence X_n is said to converge almost surely to X if

This is denoted by X_n X. Note that convergence in probability and convergence almost surely only make sense if each of X_n and X are defined on the same probability space. For convergence in distribution this is not necessary.