Let $M_n$ be a vector-valued $F_n$-martingale ($M_n:\Omega \rightarrow R^p$). Is then $\lVert M_n \rVert $ also a martingale?
I have $E(M_{n+1} | F_n)= M_n$ and liked to say something about $E(\lVert M_{n+1} \rVert \mid F_n )= \lVert M_n \rVert$.
I tried to construct a counter-example, let $M_2=Y_1+Y_2$, then it would be enough to find RV $Y_1, Y_2$ such that $E(\lVert Y_1+Y_2 \rVert \mid F_1) =\lVert Y_1 \rVert $ and $E(Y_2 \mid F_1) \neq 0$ is possible at the same time.