Page 264 - Probability and Statistical Inference

5

Concepts of Stochastic
Convergence

5.1 Introduction

The concept of the convergence of a sequence of real numbers {a ; n ≥ 1} to
n
another real number a as n → ∞ is well understood. Now, we are about to
define the notions of “convergence” of a sequence of real valued random
variables {U ; n ≥ 1} to some constant u or a random variable U, as n → ∞.
n
But, before we do so, let us adopt a minor change of notation. In the previous
2
chapters, we wrote , S and so on, because the sample size n was held
fixed. Instead, we will prefer writing and so on, in order to make the
dependence on the sample size n more explicit.
First, let us see why we need to explore the concepts of stochastic con-
vergence. We may, for example, look at or X to gather information
n:n
about the average or the record value (e.g. the average or record rainfall) in a
population. In a situation like this, we are looking at a sequence of random
variables {U ; n ≥ 1} where U = or X . Different probability calculations
n:n
n
n
would then involve n and the distribution of U . Now, what can we say about
n
the sampling distribution of U ? It turns out that an exact answer for a simple-
n
minded question like this is nearly impossible to give under full generality.
When the population was normal or gamma, for example, we provided some
exact answers throughout Chapter 4 when U = . On the other hand, the
n
distribution of X is fairly intractable for random samples from a normal or
n:n
gamma population. If the population distribution happened to be uniform, we
found the exact pdf of X in Chapter 4, but in this case the exact pdf of
n:n
becomes too complicated even when n is three or four! In situations like
these, we may want to examine the behavior of the random variables
or X , for example, when n becomes large so that we may come up with
n:n
useful approximations.
But then having to study the sequence { ; n ≥ 1} or {X ; n ≥ 1}, for
n:n
example, is not the same as studying an infinite sequence of real numbers!
The sequences { ; n ≥ 1} or {X ; n ≥ 1}, and in general {U ; n ≥ 1}, are
n
n:n
stochastic in nature.
This chapter introduces two fundamental concepts of convergence for a
sequence of real valued random variables {U ; n ≥ 1}. First, in the Section
n
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