  This paper describes sufficient conditions to ensure the correct ergodicity
of the Adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen
[Bernoulli 7 (2001) 223--242] for target distributions with a noncompact
support. The conditions ensuring a strong law of large numbers require that the
tails of the target density decay super-exponentially and have regular
contours. The result is based on the ergodicity of an auxiliary process that is
sequentially constrained to feasible adaptation sets, independent estimates of
the growth rate of the AM chain and the corresponding geometric drift
constants. The ergodicity result of the constrained process is obtained through
a modification of the approach due to Andrieu and Moulines [Ann. Appl. Probab.
16 (2006) 1462--1505].
