Branching random walk conditioned on large martingale limit

We consider a branching random walk in the non-boundary case where the additive martingale $W_n$ converges a.s. and in mean to some non-degenerate limit $W_{\mathrm{\infty }}$. We first establish the joint tail distribution of $W_{\mathrm{\infty }}$ and the global minimum of this branching random walk. Next, conditioned on the event that the minimum is atypically small or conditioned on very large $W_{\mathrm{\infty }}$, we study the branching random walk viewed from the minimum and obtain the convergence in law in the vague sense. As a byproduct, we also get the right tail of the limit of derivative martingale.