Let $X_1,...,X_n $ be i.i.d such that $X_1$~ $N(0,1)$.
Let $$T_{k,m} = \frac{(1/k)\sum_{i=1}^{k} X_{i}^2}{(1/m)\sum_{i=k+1}^{k+m} X_{i}^2}$$
where $k+m=n$.
Then $T_{k,m}$ has an $F_{k,m}$ distribution.
Here is a statement. When min{k,m} $→∞ $, the distribution of $$\sqrt{\frac{mk}{m+k}}(T_{k,m}-1)$$
can be approximated by a $N(0,2)$ distribution.
Could some please show why this statement is validated or suggest me some materials to read through? Much appreciated.