Weakly supercritical branching process in unfavourable environment (Q6548981)

Let \(\Pi:= \{Q_n; n\in\mathbb{N}\}\) be a set of i.i.d. random probability distributions on \(\mathbb{N}\cup\{0\}\) and \(Z:= \{Z_n; n\in\mathbb{N}\cup\{0\}\}\) a Bienaymé-Galton-Watson process in random environment \(\Pi\), i.e., \(Z_0=1\) and \(Z_{n+1}\) equal to the sum from \(i=1\) to \(Z_n\) of \(\xi_i^{(n)}\), \(i\in\mathbb{N}\), \(n\in\mathbb{N}\cup\{0\}\), the \(\xi_i^{(n)}\) being independent and, for fixed \(n\), distributed according to \(Q_{n+1}\). Let \(\phi_n\) be the generating function of \(Q_n\), \(\phi_1'\in(0,\infty)\), \(X_i := \ln \phi_i'(1)\), \(S_0:=0\), \(S_n:= \sum_{i=1}^n X_i\), \(i,n\in\mathbb{N}\), and suppose that \(Z\) is weakly supercritical, i.e., \(\mathbf{E}X_1> 0\) and \(\text {exists }\beta\in(0,1)\) such that \( \mathbf{E}(X_1\exp\{-\beta X_1\}) =0\).\N\NThe author investigates the asymptotic behavior of \(Y_n(1):= Z_n\), \(Y_n(t):= Z_{[nt]}\exp\{-S_{[nt]}\}\), \(t\in[0,t)\), as \(n\to\infty\), for two types of unfavorable environment:\N\begin{itemize}\N\item[(a)] \(M_n:=\max_{1\leq i\leq n}S_i< 0\) and\N\item[(b)] \(S_n\leq u\) for some positive constant \(u\).\N\end{itemize}\NSufficient conditions are given, under which the finite dimensional distributions of\N\begin{itemize}\N\item[(a)] \(\{Y_n(t), t\in[0,1] \mid M_n< 0, S_n\geq u\}\) and\N\item[(b)] \(\{Y_n(t) \mid \max\{ i : S_i = \max(0,M_n), 0\leq i\leq n\} = n, S_n\leq u\}\)\N\end{itemize}\Nconverge, as \(n\to\infty\), to\N\begin{itemize}\N\item[(a)] \(\{U_u(t),t\in[0,1]\},\) respectively,\N\item[(b)] \(\{V_u(t),t\in[0,1]\},\)\N\end{itemize}\Nboth being random processes with non-negative constant trajectories on \((0,1)\), \(\{U_u(t) > 0\}\) and \(\{V_u(t) > 0\}\) with positive probability for \(t\in(0,1)\), \(U_u(1)\) and \(V_u(1)\) taking values in \(\mathbb{N}\cup\{0\}\), \(\{U_u(1) > 0\}\) and \(\{V_u(1) > 0\}\) with positive probability.