Introduction to the Mathematical and Statistical Foundations of Econometrics

Modes of Convergence

5.3. Introduction

Toss a fair coin n times, and let Yj = 1 if the outcome of the j th tossing is heads and Yj = — 1 if the outcome involved is tails. Denote Xn = (1/n)Yl j= Yj. For the case n = 10, the left panel of Figure 6.1 displays the distribution function Fn(x)1 of Xn on the interval [—1.5, 1.5], and the right panel displays a typical plot of Xk for k = 1, 2,..., 10 based on simulated Yj’s 2

Now let us see what happens if we increase n: First, consider the case n = 100 in Figure 6.2. The distribution function Fn(x) becomes steeper for x close to zero, and Xn seems to tend towards zero.

These phenomena are even more apparent for the case n = 1000 in Figure

6.3.

What you see in Figures 6.1—6.3 is the law of large numbers: Xn = (Vn)£n = Yj ^ E[Y1] = 0 in some sense to be discussed in Sections 6.2­

6.3 and the related phenomenon that Fn(x) converges pointwise in x = 0 to the distribution function F(x) = I (x > 0) of a “random” variable X satisfying P [X = 0] = 1.

Next, let us have a closer look at the distribution function of *JnXn : Gn (x) = Fn (x Дfn) with corresponding probabilities P [лfnXn = (2k — п)Д/й], k = 0, 1,..., n and see what happens if n ^<x>. These probabilities can be displayed

1 Recall that n(Xn + 1)/2 = YTj=1(Yj + 1)/2 has a binomial (n, 1/2) distribution, and thus the distribution function Fn (x) of Xn is

Fn(x) = P[Xn < x] = P[n(Xn +1)/2 < n(x + 1)/2]

min(n,[n(x+)/2]) /

= E (k) (1/2 )n •

where [z] denotes the largest integer < z, and the sum 1= о is zero if m < 0.

2 The Yj’s have been generated as Yj = 2 • I(Uj > 0.5) — 1, where the Uj’s are random drawings from the uniform [0, 1] distribution and I(•) is the indicator function.

in the form of a histogram:

^ ґ л P [2(к — 1)/—n — —n < —nXn < 2k/—n — —n]

Hn (x) = —

2/ n

ifx e (2(k — 1)/—n — —n, 2k/—n — —n], к = 0, 1,...,n,

Hn(x) = 0 elsewhere.

Figures 6.4-6.6 compare Gn (x) with the distribution function of the standard normal distribution and Hn (x) with the standard normal density for n = 10, 100 and 1000.

What you see in the left-hand panels in Figures 6.4-6.6 is the central limit theorem:

pointwise in x, and what you see in the right-hand panels is the corresponding fact that

Gn(x + S) — Gn(x) exp[ x2/2]

lim lim = — .

s;0 n >to S 2n

The law of large numbers and the central limit theorem play a key role in statistics and econometrics. In this chapter I will review and explain these laws.

Figure 6.3. n = 1000. Left: Distribution function of Xn. Right: Plot of Xk for к = 1, 2,..., n.