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Let $\{D(t)\}_{t\geq0}$ be a subordinator, that is, a one dimensional L\'evy process with increasing sample paths. Also, the inverse subordinator $\{E(t)\}_{t\geq0}$ is defined as $$E(t):=\inf\{x\geq0:D(x)>t\},\ \ t\geq0.$$

Are the processes $\{D(t)\}_{t\geq0}$ and $\{X(E(t))\}_{t\geq0}$ independent, where $\{X(t)\}_{t\geq 0}$ is an adapted point process. If not under what conditions they are independent, say, that the condition $\{D(t)\}_{t\geq 0}$ is independent of $\{X(t)\}_{t\geq 0}$ will work?

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closed as off-topic by Shailesh, JonMark Perry, TheSimpliFire, Mostafa Ayaz, Kevin Feb 7 '18 at 9:21

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  • $\begingroup$ @Useless Well I tried the sigma field definition of independence. But was not able to prove independence. $\endgroup$ – KKK Feb 7 '18 at 0:05