4. Let X1, X2, . . . denote a sequence of random vectors with ||X,|| 0, there exists a constant Me > 0 for which P(IXAll > ME) 1 is bounded in probability. Prove the following facts for Xn, Yn E Rd. (a) If Xn = X for any random vector X, then Xn = Op(1). (b) If Xn = Op(an), then Xn = Op(an). (c) If an/bn -+ 0 and Xn = Op(an), then Xn = Op(by). (d) If Xn = Op(an) and Yn = Op(bn), then Xn + Yn = Op(max(an, br)). (e) If Xn = Op(an) and Yn = Op(bn), then X Yn - Op(anbn). If Xn = Op(an) and Yn = Op(bn), then X, Yn = Op(anbn). (f) If Xn = Op(1) and g : Red -> Rk is continuous, then g(Xn) = Op(1). (g) For d = 1, if Xn = Op(an) with an - 0 and g : R - R is continuously differentiable with g(0) = 9(0) = 0, then g(Xn) = Op(an). Show further that if g is continuously differentiable, then g(X,) = Op(an). Hint: Apply Taylor's expansion around 0. (h) For d = 1, if Var(Xn) = a,