Two limit theorems for critical Bellman-Harris branching processes (Q1060778)

Let \(\mu_ t\) be the number of particles alive at time t in a critical Bellman-Harris process. Under conditions on regular variation of the tails of the offspring distribution and lifetime distribution it is shown that for \(c_ 1<...<c_ n\leq 1\) the conditional distribution of \((\mu_{c_ 1t},...,\mu_{c_ nt})\) given \(\mu_ t>0\) exists and may be identified with the corresponding finite-dimensional distribution in a certain Markov process. The case \(n=1\), \(c_ 1=c_ n=1\) was treated earlier by \textit{V. A. Vatutin}, Teor. Veroyatn. Primen. 22, 150-155 (1977; Zbl 0391.60082).