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Consider a process $X_t=\sum_{k=1}^{\infty} \mathbf{1}_{\{T_k\le t\}}$, and a new process $\hat{X_t}=\sum_{k=1}^{\infty} z_k \mathbf{1}_{\{T_k\le t\}}$, where $P\{z_k=0\}=p, P\{z_k=1\}=1-p$, and $\{z_k\}$ are iid , and $T_k=\sum_{i}^{k}\tau_j $ ,and $P(\tau_j\le t)=1-e^{-\lambda t}$. How can I show that $X_t$ and $\hat{X_t}$ are independent.

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    They are not. $ $2011-12-12
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    do you mind explaining why? I am not too sure, thanks2011-12-12
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    Why do you think they are? What did you try to show they are? What would you try to show they are not?2011-12-12
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    Is it possible you wanted $\sum_{k=1}^\infty (1-z_k)\mathbf{1}_{\{T_k\le t\}}$ to be your definition of $X_t$?2011-12-12
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    ....if so, I think then you'd have independence.2011-12-12
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    Does this results hold for a general renewal process? see the problem here: http://math.stackexchange.com/questions/1825150/thinning-a-renewal-process-poisson-generalization2016-06-13

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