1. Let X, X1, X2, X3,... be random variables that are positive and integer valued. Show that Xn """ X if and only if for every integer k.
lim n→∞ P(Xn = k) = P(X = k)
2. Let Z1, Z2,... be iid random variables with density f . Suppose that P(Zi > 0) = 1 and that λ = limx↓0 f (x) > 0. Let Xn = n min{Z1,..., Zn}. Show that Xn """ Z where Z has an exponential distribution with mean 1/λ.