2
$\begingroup$

Let $\{X(t), t\ge 0\}$ and $\{Y(t),t\ge 0\}$ be independent Poisson processes with parameters $\lambda_1$ and $\lambda_2$, respectively. Define $Z_1(t)=X(t)+Y(t)$, $Z_2(t)=X(t)-Y(t)$, $Z_3(t)=X(t)+k$, $k$ a positive integer. Determine which of the above processes are Poisson and find $\lambda$.

Any help is appreciated! This is not homework! Thanks.

1 Answers 1

4

The last one is not Poisson since $Z_{3}(0) \neq 0$. Consider $t_1 \leq t_2 \leq t_3 \leq t_4$. Then look at $Z_{1}(t_2)-Z_{1}(t_1), Z_{1}(t_4)-Z_{1}(t_3)$ and see if they are independent. Do the same for $Z_{2}(t)$. Then you have to look at stationary increments.

  • 0
    @Didier Piau: Thank you for your effort!2011-03-07