  Lamperti's maximal branching process is revisited, with emphasis on the
description of the shape of the invariant measures in both the recurrent and
transient regimes. A truncated version of this chain is exhibited, preserving
the monotonicity of the original Lamperti chain supported by the integers. The
Brown theory of hitting times applies to the latter chain with finite
state-space, including sharp strong time to stationarity. Additional
information on these hitting time problems are drawn from the quasi-stationary
point of view.
