Set up
We consider kernel density estimation. $X_1, \ldots, X_n \overset{\mathrm{i.i.d.}}{\sim} F$(c.d.f., unknown).
$F$ has a density $f$ and then the object it wants to estimate pointwise.
Kernel Density Estimator: $$ \hat{f_n} := \frac{1}{n}\sum_{j=1}^n \frac{1}{h}K \left(\frac{y-X_j}{h} \right). $$ $h$ means bandwidth.
What I know
If $h$ satisfied $h \to 0, nh \to \infty$, then \begin{align} \mathrm{bias\ of\ }\hat{f_n}(y) &= \frac{1}{2}h^2 f''(y)\tau^2 + o(h^2),\\ \mathrm{Var}[\hat{f_n}(y)] &= \frac{1}{nh}f(y) \int K^2(y)dy + o(\frac{1}{nh}), \end{align} where $$ \tau^2 := \int y^2 K(y)dy. $$
Question
Why do the following limitation hold? $$ \frac{\mathrm{bias\ of\ }\hat{f_n}(y)}{\sqrt{\mathrm{Var}[\hat{f_n}(y)]}}\to \xi, $$ where $\xi$ is a constant.