Let the random variable Sn be defined as follows:

Sn =

⎧

⎨

⎩

0 n = 0

n k=1 Xk n ≥ 1 where Xk is the kth outcome of a Bernoulli trial such that P[Xk = 1] = p and P[Xk = −1] = q = 1 − p, and the Xk are independent and identically distributed. Consider the process {Sn : n = 1, 2,...}.

a. For what values of p (relative to q) is {Sn} a martingale?

b. For what values of p is {Sn} a submartingale?

c. For what values of p is {Sn} a supermartingale?