Cauchy distribution and stable laws.
(i) Show that o(k) – et is the characteristic function of the Cauchy distribution 1 p(x) = TI(1 + x2) by showing that p(x) = 2 100 e-ikx 4(k)dk. Hint: integration by parts – twice.
(ii) Show that the average on n independent Cauchy random variables is an identically distributed Cauchy random variable. (This means that averaging as many independent samples of such data as you like give no more precise, accurate or reliable an estimate than any single sample alone!