Download Theorem The beta(b, b) distribution converges to the normal

Theorem The beta(b, b) distribution converges to the normal distribution when b → ∞.
Proof (by Professor Robin Ryder in the CEREMADE at Université Paris Dauphine) Let
the random variable X have the beta(b, b) distribution with probability density function
fX (x) =
Γ(2b)xb−1 (1 − x)b−1
Γ(b)Γ(b)
0 < x < 1,
where b is a real, positive parameter. The mean of X is E[X] = 1/2 and the variance of
X is V [X] = 1/4(2b + 1). Substract the mean and divide by the
√ standard deviation before
taking the limit. So consider the transformation Y = g(X) = 2 2b√+ 1(X − 1/2),√which is
a one-to-one transformation from√A = {x|0 < x < 1} to B = {y| − 2b + 1 < y < 2b + 1}
with inverse X = g −1 (Y ) = X/2 2b + 1 + 1/2 and Jacobian
1
dX
= √
.
dY
2 2b + 1
Then the probability density function of Y is
1
1
y
√
fY (y) = √
fX
+
2 2b + 1
2 2b + 1 2
b−1 b−1
y
y
1
1
Γ(2b) 1
+ √
− √
= √
·
2 2 2b + 1
2 2b + 1 Γ(b)2 2 2 2b + 1
b−1
√
√
y2
Γ(2b) 1
1
−
−
·
= √
2b
+
1
<
y
<
2b + 1.
2 2b + 1 Γ(b)2 4 4(2b + 1)
p
We now apply Stirling’s formula Γ(z) = 2π/z(z/e)z (1 + O(1/z)) and get
q
b−1
2π 2b 2b 1
y2
1
1
2b
e
fY (y) = √
1+O
−
·
2π b 2b
b
4 4(2b + 1)
2 2b + 1
b
e
b−1
2b
b
1
1
y2
2
1+O
= √ √
−
· √
b
4 4(2b + 1)
2b 2b + 1 2 2π
b−1
1
y2
1
1 b−1
1+O
−
= √ 4
b
4 4(2b + 1)
2π
b−1 1
1
y2
1+O
= √
1−
2b + 1
b
2π
2
√
√
1
y
1
= √ exp −
1+O
− 2b + 1 < y < 2b + 1.
2
b
2π
Thus, in the limit b → ∞, fY (y) converges pointwise to the probability density function of a
standard normal random variable. By Scheffé’s theorem, Y converges in distribution to the
standard normal distribution.
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