Consider $Y_1, \dots, Y_n$ random variables with $n^{1/2}(\bar{Y}_{n} - \mu ) \xrightarrow{D} iid \mathcal{N}(0,\,\sigma^{2}).$
Show $g_{n}(x_n(c))=1[n^{1/2}\frac{(\bar{Y}_{n} - \mu )}{\sigma} \leq c] \xrightarrow{D} 1[Z \leq c], \quad Z \sim \mathcal{N}(0,1)$ where $c$ is some deterministic value (and $1$ the indicator function).
Any help on this (in the paper it says distributional approximation)? As far as I see I cannot apply the continuous mapping theorem. Does the sequence converge almost surely, if yes why?
Define $$X_n:=\sqrt[] {n} \frac{\overline{Y_n}-\mu} {\sigma} $$ what is given is equivalent with $X_n\stackrel{D} {\to} Z$, with $Z\sim\mathcal N (0,1)$.
The indicator function is either zero or one, so for a fixed $c$ one has:
$$P(\mathbf{1}_{\{X_n\leq c\}}=1)=P(X_n\leq c)\stackrel{}{\to} P(Z\leq c) =P(\mathbf{1}_{\{Z\leq c\}} = 1)$$ Similarly $$P(\mathbf{1}_{\{X_n\leq c\}}=0) \stackrel{}{\to} P(\mathbf{1}_{\{Z\leq c\}} = 0)$$ Hence $$\mathbf{1}_{\{X_n\leq c\}}\stackrel{D} {\to} \mathbf{1}_{\{Z\leq c\}}$$