  This article is concerned with Monte-Carlo methods for the estimation of the
trace of an implicitly given matrix $A$ whose information is only available
through matrix-vector products. Such a method approximates the trace by an
average of $N$ expressions of the form $\ww^t (A\ww)$, with random vectors
$\ww$ drawn from an appropriate distribution. We prove, discuss and experiment
with bounds on the number of realizations $N$ required in order to guarantee a
probabilistic bound on the relative error of the trace estimation upon
employing Rademacher (Hutchinson), Gaussian and uniform unit vector (with and
without replacement) probability distributions.
  In total, one necessary bound and six sufficient bounds are proved, improving
upon and extending similar estimates obtained in the seminal work of Avron and
Toledo (2011) in several dimensions. We first improve their bound on $N$ for
the Hutchinson method, dropping a term that relates to $rank(A)$ and making the
bound comparable with that for the Gaussian estimator.
  We further prove new sufficient bounds for the Hutchinson, Gaussian and the
unit vector estimators, as well as a necessary bound for the Gaussian
estimator, which depend more specifically on properties of the matrix $A$. As
such they may suggest for what type of matrices one distribution or another
provides a particularly effective or relatively ineffective stochastic
estimation method.
