Revisiting conditional limit theorems for the mortal simple branching process (Q1962610)

For standard Galton-Watson processes \(\{Z_n\}\) with offspring mean \(m\leq 1\) and extinction time \({\mathcal H}\), two of the classical limit results assert that in the subcritical case \(m<1\), \(Z_n\) has a limiting conditional distribution (the Yaglom limit) given \({\mathcal H}> n\) under weak conditions, whereas in the critical case \(m=1\) with finite offspring variance, \(Z_n/n\) has an exponential limit. The paper presents a unification and extension of a number of more general limit results related to these two: When \(m<1\), particular attention is given to the conditioning \({\mathcal H}\in n+{\mathcal S}\) where \({\mathcal S}\subset\{1,2,\dots\}\), rather than \({\mathcal H}> n\) \(({\mathcal S}= \{1,2,\dots\})\). Of particular interest is the case \({\mathcal S}= \{\nu\}\), and the limit \(\nu\to \infty\) is studied. When \(m=1\), an essentially known result on the joint conditional limit of \((Z_n,T_n)\), where \(T_n= \sum^n_1 Z_j\) is the total progeny up to time \(n\), is revisited and extended to conditionings like \({\mathcal H}<\alpha n\) or \({\mathcal H}> n+\nu_n\). The structure of the limit of \(T_n\) is studied in more detail and results of Karpenko and Nagaev on \(E[T_n\mid{\mathcal H}= n]\) are generalized.