Martingale and independent increment

Question

I know that in $L^2$ martingale a have independent increments. In particular that $\mathbb{E}[(X_m-X_n)^2]=\mathbb{E}[X^2_m-X^2_n]$ if X is a martingale. Does this extend also for general $p\geq 1$ in the form $\mathbb{E}[(X_m-X_n)^p]=\mathbb{E}[X^p_m-X^p_n]$?

Answer 1

Take a standard brownian motion $B_n$ and $p=4$.

$$\mathbb{E} \left[ (B_m - B_n)^4 \right] =\mathbb{E} \left[ B_{m - n}^4 \right] = 3 (m-n)^2 \neq 3 ( m^2 - n^2 ) = \mathbb{E} \left[ B_m^4 - B_n^4 \right]$$