A random variable has a Pareto distribution with parameters a,b (a > 0,b> 0) if its density is f(x; a,b) = a b(1 + x/b)a+1 , x> 0.

Let X1, ...,Xn be a random sample from the Pareto distribution with density f(x; 1, 1). (i) Find the limiting distribution of random variable Un = nX1:n.

[Hint: Find cdf F(x) of Xi, and then determine cdf of Un in terms of F(x).] (ii)

Show that Vn = Xn:n does not have a proper limiting distribution; specifically, limn→∞P{Vn ≤ t} = 0 for every t. (iii) Find the limiting distribution of Vn/n.