An estimate for $\beta$-Hermite ensembles via the zeros of Hermite polynomials

Let $X$ be an $N$-dimensional random vector which describes the ordered eigenvalues of a $\beta$-Hermite ensemble, and let $z$ the vector containing the ordered zeros of the Hermite poynomial $H_N$. We present an explicit estimate for $P(\|X-z\|_2\ge\epsilon)$ for small $\epsilon>0$ and large parameters $\beta$. The proof is based on a central limit theorem for these ensembles for $\beta\to\infty$ with explicit eigenvalues of the covariance matrices of the limit. The estimate is similar to previous estimates of Dette and Imhof (2009).