Approximation of an $F$ distribution when degrees of freedom tend to infinity

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.

• Too late here to think straight. Do not immediately see an intuitive argument. Simulated a million iterations with $k = m = 1000$ and the PDF of $\mathsf{Norm}(0, \sigma=\sqrt{2})$ does fit nicely on the histogram of simulated values.// The num of T converges to 1, and so does denom. – BruceET Oct 4 '17 at 7:39
• I guess a differentiable function h should be defined and use the theorem that if $\sqrt{n}(\overline{Y}-m) \stackrel{L}\rightarrow N(0,Σ)$, then $\sqrt{n}(h(\overline{Y})-h(m)) \stackrel{L}\rightarrow N(0,h^{(1)}(m)Σ[h^{(1)}(m)^{T}])$. But not sure how to do this. – Deepleeqe Oct 4 '17 at 9:50
• I'd be inclined to try to use Slutsky's theorem and then try to apply CLT to the numerator. – Glen_b Oct 10 '17 at 0:10