Non-Homogeneous Poisson Process (NPHH)

1. Properties:

• N(0) = 0
• N(t) has independent increments
• $Pr\{N(t+h) - N(t)\} = \lambda(t)h + o(h)$
• $Pr\{N(t+h)-N(t) \geq 2\} = o(h)$

Notes:

• This is like a Poisson process, without the stationary assumption

A process with the above properties is a NHPP with intensity (or rate) function $\lambda(t)$

2. Definition: 

The mean value function (for a NHPP) is

$m(t) = \int^t_0 \lambda(u) du$

3. Key Properties: 

• For a NHPP, N(s+t) - N(s) is a Poisson random variable with mean $m(s+t) - m(s)$