The problem mainly focus on stochastic convergence and law of large numbers.

probability matrix shown below. Then, state 0 is periodic. 0.5 0.5 1 0 2. (15%) A random sequence {Yn}, n = 1, 2, ..., is defined as follows: Yn = 2n X1X2 . . . Xn where X1, X2, ... are i.i.d. random variables with Xi = 0 with prob. 0.5 1 with prob. 0.5 for all i = 1, 2, .... (2) Does {Yn} converge in probability, and if so what is the limit? (22) Does {Ym } converge with probability 1, and if so what is the limit? (222) Does {Ym} converge in the mean square sense, and if so what is the limit