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 $t \geq 0$?



A periodically independent stochastic process
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 $t$.


