  The stability and ergodicity properties of two adaptive random walk
Metropolis algorithms are considered. The both algorithms adjust the scaling of
the proposal distribution continuously based on the observed acceptance
probability. Unlike the previously proposed forms of the algorithms, the
adapted scaling parameter is not constrained within a predefined compact
interval. The first algorithm is based on scale adaptation only, while the
second one incorporates also covariance adaptation. A strong law of large
numbers is shown to hold assuming that the target density is smooth enough and
has either compact support or super-exponentially decaying tails.
