  An observation of a cumulative distribution function $F$ with finite variance
is said to be contaminated according to the inflated variance model if it has a
large probability of coming from the original target distribution $F$, but a
small probability of coming from a contaminating distribution that has the same
mean and shape as $F$, though a larger variance. It is well known that in the
presence of data contamination, the ordinary sample mean looses many of its
good properties, making it preferable to use more robust estimators. From a
didactical point of view, it is insightful to see to what extent an intuitive
estimator such as the sample mean becomes less favorable in a contaminated
setting. In this paper, we investigate under which conditions the sample mean,
based on a finite number of independent observations of $F$ which are
contaminated according to the inflated variance model, is a valid estimator for
the mean of $F$. In particular, we examine to what extent this estimator is
weakly consistent for the mean of $F$ and asymptotically normal. As classical
central limit theory is generally inaccurate to cope with the asymptotic
normality in this setting, we invoke more general approximate central limit
theory as developed by Berckmoes, Lowen, and Van Casteren (2013). Our
theoretical results are illustrated by a specific example and a simulation
study.
