  For near-critical, transient Markov chains on the non-negative integers in
the Lamperti regime, where the mean drift at $x$ decays as $1/x$ as $x \to
\infty$, we quantify degree of transience via existence of moments for
conditional return times and for last exit times, assuming increments are
uniformly bounded. Our proof uses a Doob $h$-transform, for the transient
process conditioned to return, and we show that the conditioned process is also
of Lamperti type with appropriately transformed parameters. To do so, we obtain
an asymptotic expansion for the ratio of two return probabilities, evaluated at
two nearby starting points; a consequence of this is that the return
probability for the transient Lamperti process is a regularly-varying function
of the starting point.
