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10.3 Classical Orthogonal Polynomials 315
Example 10.8 (Chi-squared distribution function). For x > 0, consider calculation of the
integral
1 x 1
√ t 2 exp(−t/2) dt,
(2π) 0
which is the distribution function of the chi-squared distribution with 3 degrees of freedom.
√
Using the change-of-variable u = t,
√ x
x
1 1 2
√ t 2 exp(−t/2) dt = 2 u φ(u) du
(2π) 0 0
√
x 1
2
= 2 u φ(u) du − .
2
−∞
2
Since u = H 2 (u) − 1, using Theorem 10.8 we have that
1 x 1 √ 1 √ √
√ t 2 exp(−t/2) dt = 2 ( x) − − xφ( x) .
(2π) 0 2
It may be shown that the Hermite polynomials are complete; see, for example, Andrews,
Askey, and Roy (1999, Chapter 6). Hence, the approximation properties outlined in Theo-
rem 10.6 are valid for the Hermite polynomials.
Example 10.9 (Gram-Charlier expansion). Let p denote a density function on the real
line and let and φ denote the distribution function and density function, respectively, of
the standard normal distribution. Assume that
∞
p(x)
2
dx < ∞.
−∞ φ(x)
Under this assumption
p(x)
∞
2
d (x) < ∞
−∞ φ(x) 2
so that the function p/φ has an expansion of the form
∞
p(x)
= α j H j (x)
φ(x)
j=0
where the constants α 0 ,α 1 ,... are given by
∞
√
α j = H j (x)p(x)φ(x) dx/ ( j!);
−∞
note that α 0 = 1.
Hence, the function p has an expansion of the form
∞
p(x) = φ(x) 1 + α j H j (x) .
j=1
This is known as a Gram-Charlier expansion of the density p.In interpreting this result it is
important to keep in mind that the limiting operation refers to mean-square, not pointwise,
convergence.