  We solve the problem of asymptotic behaviour of the renewal measure (Green
function) generated by a transient Lamperti's Markov chain $X_n$ in $\mathbf
R$, that is, when the drift of the chain tends to zero at infinity. Under this
setting, the average time spent by $X_n$ in the interval $(x,x+1]$ is roughly
speaking the reciprocal of the drift and tends to infinity as $x$ grows. For
the first time we present a general approach relying in a diffusion
approximation to prove renewal theorems for Markov chains. We apply a
martingale type technique and show that the asymptotic behaviour of the renewal
measure heavily depends on the rate at which the drift vanishes. The two main
cases are distinguished, either the drift of the chain decreases as $1/x$ or
much slower than that, say as $1/x^\alpha$ for some $\alpha\in(0,1)$. The
intuition behind how the renewal measure behaves in these two cases is totally
different. While in the first case $X_n^2/n$ converges weakly to a
$\Gamma$-distribution and there is no law of large numbers available, in the
second case a strong law of large numbers holds true for $X_n^{1+\alpha}/n$ and
further normal approximation is available.
