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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.

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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.

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@PEV *Do the same for $Z_2(t)$*? Hmmm... Rather, check the sign of the variations of $Z_2$, I would say. –  Did Mar 5 '11 at 17:19
    
@Didier Piau: Can you explain more on what you said? Thanks! –  kira Mar 5 '11 at 19:30
    
@PEV: Can you also do some detailed explanation? Thanks! –  kira Mar 5 '11 at 19:33
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@kira Since the process $Z_2$ has independent increments, one must rely on another property to show that $Z_2$ fails being Poisson. For example, the fact that the random function $t\mapsto Z_2(t)$ is not almost surely nondecreasing. [Re $Z_3$, if ever this question was, after all, part of some homework of yours, you should check whether your professor imposed the condition $Z(0)=0$ for a process $(Z(t))_{t\ge0}$ to be Poisson, or not (most people do but not everybody does).] –  Did Mar 6 '11 at 20:56
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@kira: For the covariance, the only non-trivial part is the calculation of ${\rm E}[X(t) X(t+h)]$. For this purpose, write ${\rm E}[X(t)X(t + h)]$ as ${\rm E}[X(t)(X(t) + X(t + h) - X(t))]$, and note that $X(t)$ and $X(t + h) - X(t)$ are independent (further note that ${\rm E}[X(t)X(t)]$ can be found from ${\rm Var}[X(t)]$). –  Shai Covo Mar 8 '11 at 0:44
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