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.
Independence of Thinning of a Poisson process
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probability-theory
stochastic-processes
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0They are not. $ $ – 2011-12-12
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0do you mind explaining why? I am not too sure, thanks – 2011-12-12
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0Why 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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0Is 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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0....if so, I think then you'd have independence. – 2011-12-12
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0Does this results hold for a general renewal process? see the problem here: http://math.stackexchange.com/questions/1825150/thinning-a-renewal-process-poisson-generalization – 2016-06-13
1 Answers
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$X_t=0 \implies \hat{X_t}=0$
That is because you are using the same sequence of random times.
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1Perhaps you would consider expanding upon your answer? – 2011-12-13
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0What for? $ $ $ $ – 2012-01-12