From Jun Shao's Mathematical Statistics
Definition 2.10 (Consistency of point estimators). Let $X = (X_1 , ..., X_n)$ be a sample from $P ∈ \mathcal P$ and $T_n(X)$ be a point estimator of $θ$ for every $n$.
Let $\{a_n\}$ be a sequence of positive constants diverging to $∞$. $T_n(X)$ is called $a_n$-consistent for $θ$ if $a_n[T_n(X) − θ] = O_p(1)$ w.r.t. any $P∈\mathcal P$.
$O_p(\cdot)$ is defined here. When $a_n := \sqrt{n}$, the consistency is called root $n$ consistency.
Do one of $a_n$ consistency and weak consistency imply the other? if not, how about $a_n := \sqrt{n}$?
Do one of $a_n$ consistency and strong consistency imply the other? if not, how about $a_n := \sqrt{n}$?
Do one of $a_n$ consistency and $L^2$ consistency imply the other? if not, how about $a_n := \sqrt{n}$?
Thanks and regards!