Asymptotic behaviour of $\mathbb{E}[|\chi_n^2 - n|]$. I want to know how $\mathbb{E}[|\chi_n^2 - n|]$ behaves as $n\to \infty$. Simulating this in R suggest that it grows at a rate of $\sqrt{n}$, but I am unable to prove it. Setting $\chi_n^2 =\sum_{i=1}^n Z_i^2$, where $Z_i$ are iid standard normal random variables, there is a trivial upper bound
$$
\mathbb{E}[|\chi_n^2 - n]|] 
= \mathbb{E}[|\sum_{i=1}^n (Z_i^2-1)|]\\
\leq \sum_{i=1}^n \mathbb{E}[|Z_i^2 - 1|],
$$
which grows at a rate of $n$ as $n\to\infty$. However, this bound is clearly too high. What can we say about the limiting behaviour of $\mathbb{E}[|\chi_n^2 - n|]$, or is there even an explicit expression for it? 

Your simulation is correct. Let us observe that
$$
\frac1{\sqrt n}\sum_{i=1}^n(Z_i^2-1)\to\mathcal N(0,2)
$$
in distribution as $n\to\infty$ by the central limit theorem. We can show that
$$
\operatorname E\biggl|\frac1{\sqrt n}\sum_{i=1}^n(Z_i^2-1)\biggr|\to\operatorname E|Y|
$$
as $n\to\infty$, where $Z\sim\mathcal N(0,2)$ (see Theorem 25.12 and its Corollary on p. 338 of Billingsley's textbook). Hence,
$$
\operatorname E|\chi_n^2-n|=\sqrt n\operatorname E\biggl|\frac1{\sqrt n}\sum_{i=1}^n(Z_i-1)^2\biggr|\sim\sqrt n\operatorname E|Y|
$$
as $n\to\infty$.
