Almost sure convergence, convergence in probability and asymptotic normality

In the previous chapter we considered estimator of several different parameters. The hope is that as the sample size increases the estimator should get ‘closer’ to the parameter of interest. When we say closer we mean to converge. In the classical sense the sequence {xk} converges to x (xk → x), if |xk − x| → 0 as k → ∞ (or for every ε > 0, there exists an n where for all k > n, |xk − x| < ε). Of course the estimators we have considered are random, that is for every ω ∈ Ω (set of all out comes) we have an different estimate. The natural question to ask is what does convergence mean for random sequences.