Hilbert space average of transition probabilities

The typicality approach and the Hilbert space averaging method as its technical manifestation are important concepts of quantum statistical mechanics. Extensively used for expectation values we extend them in this paper to transition probabilities. In this context we also find that the transition probability of two random uniformly distributed states is connected to the spectral statistics of the considered operator. Furthermore, within our approach we are capable to consider distributions of matrix elements between states that are not orthogonal. We will demonstrate our quite general result numerically for a kicked spin chain in the integrable resp. chaotic regime.