An extension of Spitzer's integral representation theorem with an application (Q580824)

Using a new approach based on Bernstein's theorem on absolutely monotone functions, an extension is obtained of Spitzer's integral representation for the discrete branching process \(\{Z_ n;n\in {\mathbb{N}}\}\) with one- step transition probabilities given by \[ p_{ij} = \begin{cases} cp_ j^{(j)} & i=0,1,,...; \quad j=1,2,... \\ 1-c+cp_ 0^{(i)} & i=0,1,...;\quad j=0; \end{cases} \] here \(0<c\leq 1\) and \(\{p_ j^{(j)},j=0,1,...\}\) is the i-fold convolution of a probability distribution \(\{p_ j,j=0,1,...\}\) with \(0<p_ 0<1\). The result is then used to prove a complete solution to a problem in damage models satisfying a generalized Rao-Rubin condition.