Independence of Thinning of a Poisson process

Question

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.

Answer 1

$X_t=0 \implies \hat{X_t}=0$

That is because you are using the same sequence of random times.