Baire Category, Probabilistic Constructions and Convolution Squares

By an appropriate form of the Bolzano–Weierstrass theorem, we can find xj ∈
T and r(k) → ∞ such that xj(r(k)) → xj for each 1 ≤ j ≤ q. Automatically,
|xi − xj| ≥ 1/p for i = j and
q
mjxj = 0.
j=1
Since dφ(Er(k),E) → 0 it follows that xj ∈ E for 1 ≤ j ≤ q and so (E,µ) /∈
H(q,p,m) as required.
The proof that H(q,p,m) is dense forms the meat of the proof. We shall
use the simple but powerful probabilistic ideas developed in the next section.
8 The poor man’s central limit theorem
Every student learns the statement and a few students learn the proof of the
central limit theorem.
Theorem 8.1. If X1, X2, ... are independent real valued random variables
with mean 0 and variance 1, then
as n → ∞.
X1 + X2 + ... + Xn
Pr
n1/2
∈ [a,b] → 1
2π
b
a
exp(−t 2 /2)dt
However, knowing the statement, or even the proof, of a theorem is not
the same as understanding it 2 .
Exercise 8.2. (i) Quickly sketch the graph of exp x that you usually draw.
(ii) Sketch the graph of exp x as x runs from −10 to 10 paying attention
to the scales involved.
(iii) Sketch the graph of exp(−x 2 /2) as x runs from −10 to 10 paying
attention to the scales involved.
Exercise 8.2 reminds us that, if X is a random variable with a normal
distribution mean 0 and variance σ 2 , then Pr(|X| ≥ Kσ) → 0 very rapidly
as K → ∞.
2 The present author knows for certain that he did not understand the central theorem
when he was a student. He strongly suspects that he does not understand it now.
24