If I have two independent stochastic processes lets say W and N and a arbitrary stopping time $\tau_n$. Are the stopped processes $W_{\tau_n}$ and $N_{\tau_n}$ still independent in general still independent ?
2026-05-17 15:48:32.1779032912
stopped independent processes
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No, consider the case where $W$ and $N$ are continuous (e.g. independent Brownian motions with different starting values) and $\tau_n = \inf\{t : W_t = N_t\}$. Then $W_{\tau_n} = N_{\tau_n}$, so they are not independent.