# Covariance function for inhomogeneous poisson process

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!

• And your take on this would be?
– Did
Jul 21, 2017 at 4:05
• Sorry, not sure what take means?
– Drew
Jul 21, 2017 at 4:12
• 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. Jul 21, 2017 at 4:45

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