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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.
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