Extremum of a time-inhomogeneous branching random walk

Consider a time-inhomogeneous branching random walk, generated by the point process Ln which composed by two independent parts: ‘branching’ offspring Xn with the mean 1 + B(1 + n)-β for β ∈ (0, 1) and ‘displacement’ ξn with a drift A(1 + n)-2α for α ∈ (0, 1/2); where the ‘branching’ process is supercritical for B > 0 but ‘asymptotically critical’ and the drift of the ‘displacement’ ξn is strictly positive or negative for |A| > 0 but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter β and α.