show that $X_n$ convergences almost surely and in $L^2$

Question

\begin{array}{l}{\text { let } \theta \in]0,1[,\left\{\epsilon_{n}, n \geq 1\right\} \text { a sequence of Bernoulli r.v's }} \\ {\qquad X_{n+1}=\theta X_{n}+(1-\theta) \epsilon_{n+1}, \quad n \geq 0}\end{array}

Question : show that $X_n$ convergences almost surely and in $L^2$

I could show that $0 < X_n < 1$ and that $X_n$ is a martingale but I don't know what to do next.

help me please.

Answer 1

An $\mathbb L^2$-bounded martingale is Uniformly Integrable by Hölder inequality, therefore Doob's Convergence Theorem applies.

Here, the martingale is even bounded in $\mathbb L^\infty\subset\mathbb L^2$.