a question about $L^p $ convergence theorem

Question

I read the proof of $L^p$ convergence theorem of martingales but I can't exactly figure out that whether we can find a square integrable martingale that converges almost surely but not in $L^2$.

Answer 1

Let $\xi_j$ iid random variables, $\xi_j \sim \frac{1}{2} (\delta_{-1}+\delta_1)$. Then

$$X_n := \sum_{j=1}^n \xi_j \in L^2 \qquad \quad X_0 := 0$$

and $(X_n)_{n \in \mathbb{N}_0}$ is a martingale. Let $S:=\inf \{n \geq 0; X_n=-1\}$. By optional stopping we obtain that $Y_n := X_{S \wedge n}+1$ is a martingale. Since $Y_n \geq 0$ there exists

$$Y_\infty = \lim_n Y_n = X_S+1 = -1+1=0 \quad \text{a.s.}$$

But: $\mathbb{E}Y_n = \mathbb{E}Y_0=1$, $\mathbb{E}Y_\infty=0$. Hence $Y_n \not \to Y_\infty$ in $L^1$ (in particular $Y_n \not \to Y_\infty$ in $L^2$).