In this article, we consider the asymptotic behavior of three kinds
of sample breakdown points. It is shown that for the location $M$-estimator
with bounded objective function, both the addition sample breakdown point and
the simplified replacement sample breakdown point strongly converge to the
gross-error asymptotic breakdown point, whereas the replacement sample
breakdown point strongly converges to a smaller value. In addition, it is
proved that under some regularity conditions these sample breakdown points are
asymptotically normal. The addition sample breakdown point has a smaller
asymptotic variance than the simplified replacement sample breakdown point. For
the commonly used redescending $M$-estimators of location, numerical results
indicate that among the three kinds of sample breakdown points, the replacement
sample breakdown point has the largest asymptotic variance.