I'm trying to prove that the squared Brownian motion $(W_t^2)$ doesn't have independent increments, I tried using the covariance and it doesn't quite work, can anybody give me any pointer to how to prove this?
Independent Increments in Squared Brownian Motion – Brownian Motion
brownian motion
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Best Answer
Fix some $t\gt s\gt0$ and consider the event $A=[W_t^2\gt W_s^2]$. Then $\mathrm P(A\mid W_s^2=0)=1$ and $\mathrm P(A\mid W_s^2=x)\to\frac12$ when $x\to+\infty$, hence the map $x\mapsto\mathrm P(A\mid W_s^2)$ is not constant.
In particular, $W_t^2-W_s^2$ and $W_s^2$ are not independent.