  We consider a Metropolis--Hastings method with proposal $\mathcal{N}(x,
hG(x)^{-1})$, where $x$ is the current state, and study its ergodicity
properties. We show that suitable choices of $G(x)$ can change these compared
to the Random Walk Metropolis case $\mathcal{N}(x, h\Sigma)$, either for better
or worse. We find that if the proposal variance is allowed to grow unboundedly
in the tails of the distribution then geometric ergodicity can be established
when the target distribution for the algorithm has tails that are heavier than
exponential, but that the growth rate must be carefully controlled to prevent
the rejection rate approaching unity. We also illustrate that a judicious
choice of $G(x)$ can result in a geometrically ergodic chain when probability
concentrates on an ever narrower ridge in the tails, something that is not true
for the Random Walk Metropolis.
