I have a sequence of independent random variables $(X_n,n\geq 1)$ such that $X_n\sim \mathcal{E}(n)$. More precisely, $P(\{X_n\leq t\}) = 1-e^{-nt}$. From this I build the following process $(S_n,n\geq 1)$ where $S_0=0, S_n=X_1+\ldots+X_n$. In the end I have the process $M_n=S_n-E(S_n)$. I was able to show that this is a martingale, since $E(X_i)= \frac{1}{i}$, $$ E(S_{n+1}-E(S_{n+1})\mid \mathcal{F}_n)= S_n + E(X_{n+1}) - E(S_{n}) - E(X_{n+1}) = S_n - E(S_n)$$ And I have that $E(S_n)=\sum_{j=1}^n \frac{1}{j}$ and that $Var(S_n)=\sum_{j=1}^{n} \frac{1}{j^2}$ thanks to independence. I am trying to compute whether there exists a r.v. $M_{\infty}$ such that:
- $M_n\to M_{\infty}$ a.s.
- $M_n = E(M_{\infty}\mid\mathcal{F}_n)$
In order to do this I need to upper-bound $\sup_n E(M_n^2)$ or $\sup_n E(|M_n|)$ but I do not know how to proceed...any hints?
Note that the martingale $M_n := S_n-\mathbb{E}(S_n)$ satisfies
$$\mathbb{E}(M_n^2) = \text{var}(S_n) = \sum_{j=1}^n \text{var}(X_j) = \sum_{j=1}^n \frac{1}{j^2}$$
and therefore
$$\sup_{n \in \mathbb{N}} \mathbb{E}(M_n^2)<\infty,$$
i.e. $(M_n)_{n \in \mathbb{N}}$ is an $L^2$-bounded martingale. Applying a standard convergence theorem for martingales we find that there exists $M_{\infty} \in L^2$ such that $M_n \to M_{\infty}$ almost surely and in $L^2$. In particular, $M_n \to M_{\infty} \in L^1$ and from this it is not difficult to see that
$$M_n = \mathbb{E}(M_{\infty} \mid \mathcal{F}_n)$$
for all $n \in \mathbb{N}$.