I have a (possibly simple) question:
If a sequence of scalar random variables, $\{ X_T \}_{T=1}^{\infty}$, convergences in probability to a constant $c$, does that imply that the variance of $X_T$ converges to $0$? In other words, if one has an estimator $\hat{\beta}_T$ of a true parameter value $\beta_0$ with $Var(\hat{\beta}_T)$ converging to a non-zero constant, can this estimator be a consistent estimator of $\beta_0$?
Thanks in advance.