- Question:
Let $X_n$ is a martingale, which is $L^1$ bounded： $\sup_{n}E[|X_n|]=K<\infty$. Prove $$ \sum_{i=2}^\infty (X_i-X_{i-1})^2<\infty,\ a.s. $$
- Other information:
Our prof gives a hint. He let us consider a truncation. Introduce a stopping time $\tau_L:=\inf\{n:|X_n|\ge L\}$, to show $$ E[\sum_{i=2}^n (X_i-X_{i-1})^2 1_{\{\tau_L> n\}}] $$ is uniformly bounded in $n$.
- My confusion:
I do not know how to deal with the sum $\sum_{i=1}^n (X_i-X_{i-1})^2 1_{\{\tau_L> n\}}$. Do we have $ E[\sum_{i=2}^n (X_i-X_{i-1})^2 1_{\{\tau_L> n\}}]= E[(X_n-X_1)^2 1_{\{\tau_L> n\}}]\le 4L^2$? I want to show $$ E[\sum_{i=2}^\infty (X_i-X_{i-1})^2]<\infty $$