Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations

Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). We study flow of ancestral lineages as time goes past and show that, properly renormalized, it converges almost surely inverse a drift-free subordinator whose Laplace exponent is explicit in terms mechanism. The shown be partitioning current into families distinct common ancestors. When Grey’s condition satisfied, comes from discrete set ancestors have i.i.d. sizes distributed according quasi-stationary distribution CSBP conditioned on non-extinction. not continuum which described increase points S limiting subordinator. proof based general result for stochastically monotone independent interest, relates θ -invariant measures functions process its Siegmund dual.