I came across the following statement (marked as true) in multiple-choice section of an old exam:
The variance of a consistent estimator goes to zero with the growing sample size.
As far as I can tell, it can be translated as
Convergence in probability to a constant implies convergence in $L^2$.
Which is clearly false.
Is there a way to repair the statement? I mean maybe the professor forgot to mention some additional assumption typical for the context. (E.g. how convergence in probability and uniform integrability together imply $L^1$ convergence, but it seems to be irrelevant for the above statement.)