Question

Let Z1, Z2.. be a sequence of IID random variables with mean 0 and variance 1 and define i=1 and Another method of proof of CLT (the method of "moments") works by showing that for each m, the limit Lm exists, and the sequence satisfies the recurrence relation Use integration by parts to show that the sequence Rm variable, satisfies the same recurrence relation EZ"], where Z is a normal N(0, 1) random Use induction to deduce that the above recurrence relation implies Lm-Rm0 if m is odd, and that they are both equal to 2"m/2(m/2)! if m is even

Please give a detailed explain of integration by parts and the induction to prove the equation. Thank you!

Answer 1

elationi E (군m make astuHow 2.2 27, I tn) -I 2. No Uu integration 여 m-l 2 여녀2-1,-1 2

NO make athe 2. M I 2 d 2 J2ス Tr -I rm - mR

hd mte Normal distri buon m e odd En- emue one even

Lh 2ー Th (nt)(m +) m (mt2) 2- m12) 2 , 2逕丿