I'm working on the following problem: Let $X$ be a zero mean stochastic process with independent and stationary increments. I want to prove that the function $t \mapsto \mathbb{E}X_t^2$ is linear. I have already proven that $X$ and $(X_t^2-\mathbb{E}X_t^2)_t$ are martingales, maybe this can be used.

My attempt: By stationary increments we have that $X_{t+s}-X_s \overset{d}{=} X_t - X_0$, squaring and shuffling gives $X_{t+s}^2 \overset{d}{=} X_t^2+2X_sX_{t+s}-X_s^2+X_0^2-2X_0X_t$. Taking expectations I then get $\mathbb{E}X_{t+s}^2 = \mathbb{E}X_t^2+\mathbb{E}X_s^2+\mathbb{E}(X_0^2-2X_0X_t)$. However I'm stuck with this last term which I can't see being zero. Furthermore I have no start when trying to show that $\mathbb{E}X_{ct}^2=c\mathbb{E}X_t^2$.

Thanx in advance!


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