Note that $a$ has a mean of 0.
My approach:
$$X_t=X_{t-1}+a_t$$ $$E[X_{t+1}|X_1 + \dots+X_{t-1}]$$$$E[X_{t+1}\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}+2a|X_1 + \dots+X_{t-1}]$$$$=E[X_{t-1}+2a\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}|X_1 + \dots+X_{t-1}]+E[2a|X_1 + \dots+X_{t-1}]$$$$=E[X_{t-1}\mid X_1 + \dots+X_{t-1}]+E[2a\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}|X_1 + \dots+X_{t-1}]+0$$$$=E[X_{t-1}\mid X_1 + \dots+X_{t-1}]+0$$ $$=E[X_{t-1}|X_1 + \dots+X_{t-1}]$$$$=E[X_{t-1}\mid X_1 + \dots+X_{t-1}]$$ $$=X_{t-1}$$ Am I doing something wrong here? shouldn't the end product be $X_{t}$$X_t$?
