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If we have an inhomogeneous Poisson process with intensity $\lambda(t)$, what does the covariance function $\mathbb{E}[X_s, X_t]$ look like? Can anyone point me to a derivation?

I would like to ask the same question for a Hawkes type process, where the intensity can be "level-dependent".

If these question have a complicated answer, what I am really interested in is that I have an entire covariance matrix $\Sigma(t, s)$ for a set of discrete $t$ and $s$ and I would like to use it to estimate $\lambda(t)$.

Thank you!

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  • $\begingroup$ And your take on this would be? $\endgroup$
    – Did
    Jul 21, 2017 at 4:05
  • $\begingroup$ Sorry, not sure what take means? $\endgroup$
    – Drew
    Jul 21, 2017 at 4:12
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    $\begingroup$ It means: Please show how much you understand and exactly where you are having trouble. What are your attempts, thoughts, and ideas? People don't like solving other's problems from scratch. $\endgroup$
    – Graham Kemp
    Jul 21, 2017 at 4:45

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For an inhomogeneous Poisson process with intensity $\lambda(t)$, $X(t)$ (representing the number of occurrences in the interval $[0, t]$) is a Poisson random variable with parameter $\Lambda(t) = \int_0^x \lambda(x)\; dx$. The covariance is $\text{Cov}(X_s, X_t)$. But the numbers of occurrences in disjoint intervals are independent. Thus if $s \le t$, $X(s)$ and $X(t) - X(s)$ are independent, and

$$\text{Cov}(X(s), X(t)) = \text{Cov}(X(s), X(s)) + \text{Cov}(X(s),X(t) - X(s)) = \text{Cov}(X(s),X(s)) = \text{Var}(X(s)) = \Lambda(s)$$

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