5. (20 marks) (1) Given that {Mn} is a martingale and f() is a convex function, show that {f(Mn)} is a sub-martingale. (2) Suppose that X1,X2, are i.i.d random variable with expected value =EXi ( is finite), and variance Var(Xi)<. define mn="nX1X2Xn." is the natural logarithmic function a martingale sub-martingale or sup-martingale your reason should be as explicit possible. hint: jensen inequality: if f convex then e provided expectations exist where x random variable. for>