Solution
= Solution
The <central limit theorem> gives
$$
\sqrt n(\overline X_n-\mu)\xrightarrow{d}N(0,1).
$$
Applying the <delta method> to $g(x)=x^2$ yields
$$
\boxed{
\sqrt n(T_n-\mu^2)
\xrightarrow{d}N(0,4\mu^2)
}.
$$
For $\mu=0$ this denotes the degenerate distribution at zero, which is the correct limit at the $\sqrt n$ scale.