Autocorrelation sum of poisson

Question

Let $x$ and $y$ be independent poisson random variables with parameters $\lambda_1$ and $\lambda_2$. Let $Z=x+y$. What is the autocorrelation for $Z$ in $t_1$ and $t_2$, i.e., what is $R_Z(t_1,t_2)$?

Answer 1

Your question is very unclear. But here's an attempt:

Lets start with the autocovariance, which is

$$cov(Z_{t_1}, Z_{t_2}) = cov (X_{t_1} + Y_{t_1}, X_{t_2} + Y_{t_2}) \qquad (\ast)$$

Which gives:

$$(\ast) = cov(X_{t_1}, X_{t_2}) + cov(X_{t_1}, Y_{t_2}) + cov(Y_{t_1}, Y_{t_2}) + cov(Y_{t_1}, X_{t_2})$$

Since $X$'s and $Y$'s are independent, we have $\mathbb{E}(X_{t}Y_{s}) = \mathbb{E}(X_{t})\mathbb{E}(Y_{s})$, so that leaves us with:

$$(\ast) = cov(X_{t_1}, X_{t_2}) + cov(Y_{t_1}, Y_{t_2}) = \mathbb{E}(X_{t_1}X_{t_2}) - \lambda_1^2 + \mathbb{E}(Y_{t_1}Y_{t_2}) - \lambda_2^2$$

Now, if we assume these are weakly stationary, then we get:

$$\gamma_X(t_2 - t_1) - \lambda_1^2 + \gamma_Y(t_2 - t_1) - \lambda_2^2 $$

where $\gamma_X$ and $\gamma_Y$ are autocovariance functions of $X$ and $Y$.