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Theorem The limiting distribution of the gamma(α, β) distribution is the N (µ, σ 2 ) distribution where µ = αβ and σ 2 = α2 β.
Proof Let the random variable X have the gamma(α, β) distribution with probability density
function
x β−1 e−x/α
fX (x) =
x > 0.
αβ Γ (β)
The moment generating function of X is
MX (t) = (1 − αt)−β
t<
1
.
α
The mean of X is E[X] = αβ and the variance of X is V [X] = α2 β. Subtract the mean and
divide by the standard deviation before taking the limit. The transformation
Y = g(X) = (X − αβ)
. q α β =X
. q α β −
q
β
√
is a 1–1 transformation from X = {x | x > 0} to Y = {y | y > − β}. The moment generating
function of Y is
h
MY (t) = E etY
=
i
 . √ √ 
α β − β
t X

E
e

 . √ 
α β
t X

β
E

e
= e−t
√
= e−t
√
= e
−t
β
√ β
. q α β
MX t
.q −β
1− t
β
t<
q
β.
The limiting moment generating function of Y is
lim MY (t) = lim e
β→∞
β→∞
−t
√ β
.q −β
1− t
β
= et
2 /2
− ∞ < t < ∞,
which is the moment generating function of a standard normal random variable. This limit
can be found using the maple statement
limit(exp(-t * sqrt(beta)) * (1 - (t / sqrt(beta))) ^ (-beta), beta = infinity);
which confirms that the limiting distribution of the gamma distribution as β → ∞ is the
normal distribution.
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