Extremum of a time-inhomogeneous branching random walk

摘要

Consider a time-inhomogeneous branching random walk, generated by the point process L-n which composed by two independent parts: 'branching' offspring X-n with the mean 1 + B(1 + n)(-beta) for beta is an element of (0, 1) and 'displacement' xi(n) with a drift A(1 + n)(-2 alpha) for alpha is an element of (0, 1/2); where the 'branching' process is supercritical for B 0 but 'asymptotically critical' and the drift of the 'displacement' xi(n) 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 beta and alpha.