Can the Adaptive Metropolis Algorithm Collapse without the Covariance Lower Bound?

that is, the sample covariance matrix of the history of the chain plus a (small) constant ǫ > 0 multiple of the identity matrix I. The lower bound on the eigenvalues of Sn induced by the factor ǫI is theoretically convenient, but practically cumbersome, as a good value for the parameter ǫ may not always be easy to choose. This article considers variants of the AM algorithm that do not explicitly bound the eigenvalues of Sn away from zero. The behaviour of Sn is studied in detail, indicating that the eigenvalues of Sn do not tend to collapse to zero in general. In dimension one, it is shown that Sn is bounded away from zero if the logarithmic target density is uniformly continuous. For a modification of the AM algorithm including an additional fixed component in the proposal distribution, the eigenvalues of Sn are shown to stay away from zero with a practically non-restrictive condition. This result implies a strong law of large numbers for super-exponentially decaying target distributions with regular contours.