Is the running average a martingale?

Question

$\xi_1, \ldots, \xi_n, \ldots$ - summable random variables, for every natural n: $\mathcal F_n = \sigma(\xi_1, \ldots, \xi_n)$ and $\xi_1 + \ldots + \xi_n \over n$ = $\mathbb E(\xi_{n+1}|\mathcal F_n)$. Prove that this sequence: $X_n \triangleq \frac {\xi_1 + \ldots + \xi_n} n$ is a martingale.

Answer 1

Clearly $X_{n}$ is $\mathcal{F}_{n}$-measurable and $E\left[|X_{n}|\right]<\infty$. We go to show that $E\left[X_{n+1}\mid\mathcal{F}_{n}\right]=X_{n}$. By direct calculation, \begin{eqnarray*} E\left[X_{n+1}\mid\mathcal{F}_{n}\right] & = & E\left[\frac{\xi_{n+1}}{n+1}\mid\mathcal{F}_{n}\right]+E\left[\frac{\xi_{1}+\ldots+\xi_{n}}{n+1}\mid\mathcal{F}_{n}\right]\\ & = & \frac{1}{n+1}\cdot E\left[\xi_{n+1}\mid\mathcal{F}_{n}\right]+\frac{\xi_{1}+\ldots+\xi_{n}}{n+1}\\ & = & \frac{1}{n+1}\cdot\frac{\xi_{1}+\ldots+\xi_{n}}{n}+\frac{\xi_{1}+\ldots+\xi_{n}}{n+1}\\ & = & \frac{\xi_{1}+\ldots+\xi_{n}}{n}\left\{ \frac{1}{n+1}+\frac{n}{n+1}\right\} \\ & = & X_{n}. \end{eqnarray*} Therefore $\{X_{n},\,\,n\in\mathbb{N}\}$ is a martingale with respect to the filtrarion $\{\mathcal{F}_{n},\,\,n\in\mathbb{N}\}$.