# Chi squared distribution N(0,12)N(0,\frac{1}{2}) asymptotically distributed

How can one proof the following:

Let SnS_n be χ2n\chi_n^2-distributed. Then √Sn−√n\sqrt{S_n}-\sqrt{n} is asymptotically N(0,12)N(0,\frac{1}{2}) distributed, i.e. √Sn−√n\sqrt{S_n}-\sqrt{n} ~. N(0,12)N(0,\frac{1}{2}).

I wanted to use the follwing result: If √n(Tn−n)\sqrt{n}(T_n-n) ~. N(0,σ2)N(0, \sigma^2), then for a differentiable function hh with h(μ)≠0h(\mu) \neq 0 we get

√n(h(Tn)−h(μ))\sqrt{n}(h(T_n)-h(\mu)) ~. N(0,σ2h′(μ)2),N(0,\sigma^2 h'(\mu)^2),

but I don’t know how to start. Can someone give me hint or help me?

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would it help that SnS_n ~Gamma(n/2,1/2) Gamma(n/2,1/2)? That can be proved using the mgf of the gamma distribution after having found that the square of a standard normal is gamma.
– Max Freiburghaus
2 days ago

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