Asymptotic distribution

Let $d\ge 1$ be fixed, and equip ${\mathbb{R}}^d$ with the Borel sigma-algebra generated by its Euclidean open sets. Let $Z_n$ be measurable random elements of this Borel space, not necessarily defined on a common probability space, and let $\nu$ be a Borel probability measure on ${\mathbb{R}}^d$. The sequence $(Z_n)$ has asymptotic distribution $\nu$ if its laws converge weakly to $\nu$; equivalently, $Z_n\Rightarrow Z$ for any random vector $Z$ with law $\nu$ (Convergence in distribution of random elements, Weak convergence of borel probability measures).

For an estimator, any centering or scaling, such as $r_n(T_n-\psi (\theta ))$, is part of the sequence whose asymptotic distribution is being specified; it is not implicit in the term. Weak convergence alone does not assert convergence of moments.