Show that X n is a discrete-time martingale with respect to Y if E[Y i ] = 0. How about if E[Y i ] ≥ 0? Is X n a martingale with respect to itself? What can you say about X` n if the Y i are positive...

Show that X_n
is a discrete-time martingale with respect to Y if E[Y_i] = 0. How about if E[Y_i] ≥ 0? Is X_n
a martingale with respect to itself? What can you say about X` n if the Y_i
are positive with E[Y_i] = 1? or ≥ 1?