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Consider a Poison a process $N = (N_t )_{t\geq 0}$ of intensity $\lambda >0$ whose instants of jumps are $(T_n)_{n\geq0} $ $(T_0 =0)$ and a process $\tilde{N} =(\tilde N_t )_{t\geq 0}$ defined as follows

$$ \tilde {N_t} := \left( \sum _{k=1}^{N_t} \mathbf {1}_{A_k} \right) \ \mathbf {1}_{\{ N_t \geq 1\}} $$

where $\{A_k\}_{k\geq 0} $ are independent events of identical probability $p \ (0<p<1)$, also independent of the $(T_{n+1}- T_n )_{n \geq0}$.

I am interested in calculate $\mathbb P \{ \tilde N_t =m \ |\ N_t =n \}$ but I'm stucked . Obviously it's $0$ if $n<m$, otherwise we must have

$$\mathbb P \{ \tilde N_t =m \ |\ N_t =n \}= \mathbb P \left\{ \left( \sum _{k=1}^{n} \mathbf {1}_{A_k} \right) \ \mathbf {1}_{\{ n \geq 1\}} =m \ , \ N_t =n \right\}/ \mathbb P\{ N_t =n\} = \mathbb P \left\{ \left( \sum _{k=1}^{n} \mathbf {1}_{A_k} \right) \ \mathbf {1}_{\{ n \geq 1\}} =m \ , T_n \leq t <T_{n+1} \right\} \frac{n!}{(\lambda t)^n} \exp(\lambda t) $$

I know $\mathbb P \{ \left( \sum _{k=1}^{n} \mathbf {1}_{A_k} \right) \ \mathbf {1}_{\{ n \geq 1\}} =m \}=C_{n, m} p^m (1-p)^{m-n}$ if $n\geq n $ and $0$ otherwise. As you may see I am stuked without knowing how to manage the independence of $(A_k)$ and $(T_{n+1}- T_n)$ in order to finish the calculation.

Any advice is appreciated. Thanks in advance.

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"Any advice is appreciated" Are you sure about that? –  Did Oct 28 '13 at 21:10
    
@Did: Totally! I really don't got what was the misunderstanding before … Even then I excuse for any inconvenient. –  Paul Oct 28 '13 at 21:27

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