Two Proofs of the Central Limit Theorem
Yuval Filmus
January/February 2010
In this lecture, we describe two proofs of a central theorem of mathemat-
ics, namely the central limit theorem. One will be using cumulants, and the
other using moments. Actually, our proofs won’t be entirely formal, but we
will explain how to make them formal.
1 Central Limit Theorem
What it the central limit theorem? The theorem says that under rather gen-
eral circumstances, if you sum independent random variables and normalize
them accordingly, then at the limit (when you sum lots of them) you’ll get a
normal distribution.
For reference, here is the density of the normal distribution N(µ, σ
2
) with
mean µ and variance σ
2
:
1
√
2πσ
2
e
−
(x−µ)
2
2σ
2
.
We now state a very weak form of the central limit theorem. Suppose that
X
i
are independent, identically distributed random variables with zero mean
and variance σ
2
. Then
X
1
+ ··· + X
n
√
n
−→ N(0, σ
2
).
Note that if the variables do not have zero mean, we can always normalize
them by subtracting the expectation from them.
The meaning of Y
n
−→ Y is as follows: for each interval [a, b],
Pr[a ≤ Y
n
≤ b] −→ Pr[a ≤ Y ≤ b].
1
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