1) Theorem:

An asymptotically unbiased estimator 'theta hat' for 'theta' is a consistent estimator of 'theta' IF

lim Var(theta hat) = 0

n->inf

Now my question is, if the limit is NOT zero, can we conclude that the estimator is NOT consistent? (i.e. is the theorem actually "if and only if", or is the theorem just one way?)

2)

I'm OK with part a, but I am stuck badly in part b. The only theorem I learned about consistency is the one above. Using the theorem, how can we prove consistency or inconsistency of each of the two estimators? I am having trouble computing and simplifying the variances...

Thank you for your help!