# The auto-correlation of poissonian point process

#### humenghu

##### New Member
I saw in a paper that the autocorrelation function of a poissonian point process is the following, R(t)=v^2+v*delta(t), written in Matlab style. The t is the lag of the autocorrelation, v is the intensity of the (homogeneous)poissonian process and delta is the delta function. My first question is how to prove it.

I was trying to follow a proof of this at http://icwww.epfl.ch/~gerstner/SPNM/node34.html#SECTION02425100000000000000

but I did not figure out where coefficient 1/2 1/3 1/4 etc are from, for the summation in Equ. 5.37. It was obvious though that this would give a Taylor expansion of an exponential.

Please also show me any alternative proofs.

My second question is, what is the the autocorrelation function of the inhomogeneous poisson point process, and how to prove it.

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#### humenghu

##### New Member
I figured out the answer of the first problem already. Here is the hint. For example, assume there is a spike at time t, we want to know the probability of observing three subsequent spikes at the time t+ss.t+ss+sss, t+s-sss-ss. Note that the interval between the first and forth spike is s which is a constant here. Next,assume that sss' is a possible sample value of a random variable sss, we can just convolve the p(ss)p(s-sss'-ss) dss over the interval (0,s-sss') and time the result with the p(sss'). The final step is to evaluate this over the interval (0,s) for sss'. The result will be v^3*exp(-v*s)*s^2/2. v is the intensity of the poisson process.

However, I still hope someone can help me with the other question I raised, the autocorrelation of the inhomogeneous poisson process.

#### yerti

##### New Member
Do you have a solution for the autocorrelation of the inhomogeneous poisson process? I also need help with that.