\(L^p\)-norms of Hermite polynomials and an extremal problem on Wiener chaos
The author establishes sharp asymptotics for the \(L^p\) norm of Hermite polynomials of degree \(n\) as \(n\) goes to \(\infty\), using asymptotic expansions obtained by \textit{M. Plancherel} and \textit{W. Rotach} in [Comment. Math. Helv. 1, 227-254 (1929; JFM 55.0799.02)] for such polynomials, and deduces a convergence result in distribution for Wick powers of order \(n\) as \(n\) goes to \(\infty\). In [``Gaussian Hilbert spaces'' (1997; Zbl 0887.60009)], \textit{S. Janson} has noted that for all even integer values of \(p\) the functional \(\|X\|_p/\|X\|_2\) is maximized in the homogeneous Wiener chaos \(H^{:n:}\) by Wick powers of order \(n\), and has asked whether this result holds for other values of \(p\). In the paper under review the author provides a negative answer for all \(0<p<2\) and all sufficiently large \(n\), when \(H\) is an infinite-dimensional Gaussian Hilbert space. A conjecture stating that the property is also false for all \(n\geq 2\) is backed by numerical calculations.
scientific article; zbMATH DE number 1928878