Large families and exceedances in branching processes

We review some recent results on exceedances and genealogy of branching processes. Define a family in a branching population as a subpopulation of individuals living at time t and having a common ancestor at time τ,τ < t. We present limit theorems for the number of large families when both t and t tend to infinity. We also show that if τ =0 and the underline branching process starts with large number of ancestors, then the large family process is approximated in the Skorohod topology by processes with independent increments. We also present limit theorems for the index of the first process in the sequence of branching processes which exceeds a threshold.