Error bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models

This paper gives the Kolmogorov and Wasserstein bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models. The Kolmogorov bound is new while the Wasserstein bound improves a result obtained recently by Kirkpatrick and Nawaz [Journal of Statistical Physics, $\mathbf{\text{165}}$ (2016), no. 6, 1114–1140]. The proof is based on Stein’s method for exchangeable pairs.