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I am working on a problem and I encountered the following situation:

$(N(t): t \ge 0)$ is a Poisson process with parameter $\lambda t $. If $T_{n} = \sum_{i=1}^n W_i$ represents the $n^\text{th}$ arrival time, $W_{i}$ being inter-arrival times, then at time $T_1$, the value of $N(T_1) = 1$.

If $(G(t): t \ge 0)$ is a gamma process with parameter $ \alpha t$ and $\beta$, then $N(G(t))$ is a Poisson-Gamma subordinated process. At time $T_n$, what is the value of $N(G(T_n))$ ?

As I understand, since the Poisson process is directed by a subordinated process, the value of $N(G(T_n))$ is certainly not equal to $n$.

Could I get some help or suggestions with regards to the above question ? Any leads would be highly appreciated.

Thanks in advance.

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    $\begingroup$ What exactly is your question? $N(G(T_n))$ will of course not be deterministic, so what do you mean by "what is the value of $N(G(T_n))$"? $\endgroup$
    – unwissen
    Commented Jul 13 at 14:51
  • $\begingroup$ The count $N(T_{1})$ is 1 in the first time period since $T_{1}$ is the first arrival time of the Poisson process. For the subordinated process $ N(G(T_{1})) $, $G(T_{1})$ transforms the time scale and it affects the distribution of the inter-arrival times but does it affect the count in the first time period or will it still remain 1 ? What would be the count in $n$ time periods ? Basically I would like to know the intricacies of choosing a subordinated process as a counting process. $\endgroup$
    – Rosy
    Commented Jul 15 at 13:00
  • $\begingroup$ If you mean $N(t) \sim \operatorname{Poisson}(\lambda t)$ and that there are independent increments, then I'd call $\lambda$ rather than $\lambda t$ the parameter. Or the "rate" or the "intensity." $\endgroup$
    – Michael Hardy
    Commented Aug 25 at 20:01

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