Average of linearly correlated zero mean and unit variance random variables forms a martingale process?

Question

Suppose random variables $X_1, X_2, \dots$ are all linear correlated with value $\rho$. Suppose further that all these random variables are zero mean and has unit variance. Define the stochastic process $Y_n = \frac{1}{n}\sum_{i=1}^n X_i$. Is the stochastic process $(Y_t)$ a martingale? That is, does the following hold? $$ E[Y_{n+1} | Y_n] = Y_n $$

My intuition tells me this is true but I am unable to derive this result.

Answer 1

If $X$ is, say, $N(0,1)$, and $X_n=X$ for all $n$ then $E(Y_{n+1}|Y_n)=Y_{n+1}\neq Y_n$.