Page 265 - Probability and Statistical Inference

242 5. Concepts of Stochastic Convergence

5.2, we discuss the notion of the convergence in probability (denoted by )
and the weak law of large numbers (WLLN). We start with a weak version of
the WLLN, referred to as the Weak WLLN, followed by a stronger version,
referred to as Khinchine’s WLLN. In the Section 5.3, we discuss the notion
of the convergence in distribution or law (denoted by ). We provide Slutsky’s
Theorem in Section 5.3 which sets some of the ground rules for manipula-
tions involving these modes of convergence. The central limit theorem (CLT)
is discussed in Section 5.3 for both the sample mean and sample variance
. This chapter ends (Section 5.4) with some interesting large-sample prop-
erties of the Chi-square, t, and F distributions.
5.2 Convergence in Probability

Definition 5.2.1 Consider a sequence of real valued random variables {U ; n
n
≥ 1}. Then, U is said to converge in probability to a real number u as n → ∞,
n
denoted by , if and only if the following condition holds:

In other words, means this: the probability that U will stay away
n
from u, even by a small margin e, can be made arbitrarily small for large
enough n, that is for all n ≥ n for some n ≡ n (ε). The readers are familiar
0
0
0
with the notion of convergence of a sequence of real numbers {a ; n ³ 1} to
n
another real number a, as n → ∞. In (5.2.1), for some fixed ε (> 0), let us
denote which turns out to be a non-negative real
number. In order for U to converge to u in probability, all we ask is that p (ε)
n
n
> 0 as n → ∞, for all fixed ε(> 0).
Next we state another definition followed by some important results. Then,
we give a couple of examples.
Definition 5.2.2 A sequence of real valued random variables {U ; n ≥ 1} is
n
said to converge to another real valued random variable U in probability as n
→ ∞ if and only if as n → ∞.
Now, we move to prove what is known as the weak law of large numbers
(WLLN). We note that different versions of WLLN are available. Let us begin
with one of the simplest versions of the WLLN. Later, we introduce a stron-
ger version of the same result. To set these two weak laws of large numbers
apart, the first one is referred to as the Weak WLLN.
Theorem 5.2.1 (Weak WLLN) Let X , ..., X be iid real valued
n
1
random variables with E(X ) = µ and V(X ) = σ , −∞ < µ < ∞,0 <
2
1 1