Asymptotic Methods in Statistical Inference - Statistics Centre

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4. Delta method; Liapounov’s Theorem
• Hermite polynomials: Let = Φ 0 be the (0 1)
density, note 0 () =−(); continuing we obtain
() () =(−1) ()() (4.1)
where () is the Hermite polynomial:
1 () = 2 () = 2 −1 3 () = 3 −3 .
Differentiating both sides of (4.1) gives
+1 () = () − 0 ()
Note that (integrating by parts repeatedly)
Z ∞
−∞ (−1) ()()
Z ∞
=
−∞ () Z ∞
() =(−)
−∞ (−1) ()
= =(−) Z ∞
= (−) ()
−∞ ()