Below is actually a statement in textbook. But I don't have a good intuition of it.


If we want a stochastic process $W\_t$ to satisfy
i). $s\neq t$ implies $W\_s$ and $W\_t$ are independent,
ii). $\{W\_t\}$ is stationary,
iii). $E[W\_t]=0$ for all t,
then $W\_t$ cannot have continuous paths.


I hope someone can point out the essence of this argument to me. Also, Is there a continuous process satisfying the first two requirements?


Thanks!



Non-existence of such a continuous stochastic process
Something much stronger holds. One can actually show that no nontrivial such process has measurable sample parths. No assumption on the mean and no stationarity assumption is necessary. This is Proposition 1.1. in Y. Sun, [The almost equivalence of pairwise and mutual
independence and the duality with exchangeability](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.42.2689&rep=rep1&type=pdf), Probab. Theory Relat. Fields 112, 425- 456 (1998). The proof is not very complicated.


In terms of the intuition, continuity tells you that you can understand the behavior at a point by observing points close by- independence tells you you cannot learn anything from such points.


