Question

2. This is a more convincing case that convergence in law may not imply convergence of density functions. Let fo be a pdf which is symmetric about zero, and denote its CF by фо(t). Convince yourself that po is real-valued. (The converse is true.) (i) Show that (1-cos(nr)%(r) , 0snar)ola) 1-eo(n) fn(x)-- (n 1,2, . ..), is a pdf. Explain why the sequence (n) does not converge, with the possible exception of isolated points (i) Show that the CF of fn is m(t)900-n+t 1-Po(n) (ii) For n 20 let Yn be a random variable with CF n. Use the continuity theorem for CF's to show that Y,与 You have to use the Riemann-Lebesgue lemima, and you can find this on the Internet (Wikipedia) https: //en. wikipedia.org/wiki/Riemann%E2%80%93 Lebesgue-lemma

Answer 1