Exact behavior of Gaussian seminorms
Let \(\{x_i\}^\infty_{i = 1}\) be a sequence of independent standard Gaussian random variables. Consider the random variable \(z = \sum^\infty_{i = 1} x^2_i/a^2_i\), where \(\{a_i\}^\infty_{i = 1}\) is a deterministic sequence such that \(\sum^\infty_{i = 1} a^2_i < \infty\). The authors show that one may find an explicit function \(f(\varepsilon)\) such that \(\lim_{\varepsilon \to 0} f(\varepsilon) P\{z < \varepsilon\} = 1\).
scientific article; zbMATH DE number 768777