Let, Xk k = 1, 2, 3… be a sequence of IID Cauchy random variables with
and let Sn be the sequence of sample means,
(a) Show that also follows a Cauchy distribution.
(b) Prove that in this case, the sample mean does not converge in probability and therefore the weak law of large numbers does not apply. What assumption has been violated in this case that makes the weak law of large numbers not applicable?