We observe a series of random i.i.d. variables, $X_1$, $X_2$, etc., all with the same mean $M$.
If we define $A_n$ as the average of the first $n$ observations, then we know, by the law of large numbers, that it converges to $M$ as n goes to infinity.
But what if we keep a moving average (correct me if this is not the correct term), according to this formulas:
$A_1 = X_1$
$A_{n+1} = p A_n + (1-p) X_{n+1}$
Where:
$0 < p < 1$
Intuitively, it seems that $A_n$ will also converge to $M$, but I couldn't prove or disprove this.