Can we say $Y_n$ is a martingale if the expectation is a constant?

Question

For example, we have $P(X_i=1)=0.5,P(X_i=-1)=0.5,(i\ge 0)$, $Y_n=X_nX_{n-1},(i\ge 1)$. I want to examine whether $Y_n,(i\ge 1)$ is a martingale.

My thinking are as follows: $E[Y_{n+1}|X_1,...,X_n]=E[X_{n+1}X_n|X_1,...,X_n]=X_nE[X_{n+1}|X_1,...,X_n]=X_nE[X_{n+1}]=0$. It's a constant. How can I say whether $Y_n,(i\ge 1)$ is a martingale?

Answer 1

We are asked to find out if $E\{Y_{n+1}|Y_1,Y_2,...,Y_n\}=Y_n$. This is same as $E\{X_{n+1}X_n|Y_1,Y_2,...,Y_n\}=Y_n$. To see that this is not true note that if $A=\{Y_n >0\}$ then $\int_A Y_n \, dP>0$ but $\int_A X_{n+1} X_n dP=\int X_{n+1} dP \int_A X_n dP=0$. Hence $\{Y_n\}$ is not a martingale.