Does there exist a non constant almost surely continuous stochastic process $X$ on $[0, \infty)$ with $X_t$ independent of $X_{t+1}$ for all $t \geq 0$?
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Stupid answer: the trivial process $X_t=0$.
Less stupid answer: for every half-integer $n/2$, choose $X_{n/2}$ independently according to your favorite probability distribution and then interpolate linearly for other values of $t$.
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$\begingroup$ Ah sorry I forgot to add nonconstant in the problem statement. Also yeah I suppose that works! $\endgroup$ Commented Sep 26, 2021 at 5:04