  The Adaptive Metropolis (AM) algorithm is based on the symmetric random-walk
Metropolis algorithm. The proposal distribution has the following
time-dependent covariance matrix at step $n+1$ \[
  S_n = Cov(X_1,...,X_n) + \epsilon I, \] that is, the sample covariance matrix
of the history of the chain plus a (small) constant $\epsilon>0$ multiple of
the identity matrix $I$. The lower bound on the eigenvalues of $S_n$ induced by
the factor $\epsilon I$ is theoretically convenient, but practically
cumbersome, as a good value for the parameter $\epsilon$ may not always be easy
to choose. This article considers variants of the AM algorithm that do not
explicitly bound the eigenvalues of $S_n$ away from zero. The behaviour of
$S_n$ is studied in detail, indicating that the eigenvalues of $S_n$ do not
tend to collapse to zero in general.
