Gaussian measures of large balls in \({\mathbb{R}}^ n\). (English) Zbl 0727.60011
Stable processes and related topics, Sel. Pap. Workshop, Ithaca/NY (USA) 1990, Prog. Probab. 25, 1-25 (1991).
[For the entire collection see Zbl 0718.00011.]
The author investigates the asymptotic behaviour of the function \[ u\to \mu \{x\in R^ n:\;\| x-x_ 0\| >u\}\quad as\quad u\to \infty, \] where \(x_ 0\in R^ n\), \(\mu\) is a symmetric Gaussian measure on \(R^ n\) and \(\| \cdot \|\) is a norm possessing some special properties. For this purpose Laplace’s method is extended to the integral \(\int_{K}e^{-u^ 2h(x)+ug(x)}dx\) for some functions h and g on a compact set \(K\subset R^ n\). The general results then are applied to \(l_ p\)-norms, \(p>0\).
The author investigates the asymptotic behaviour of the function \[ u\to \mu \{x\in R^ n:\;\| x-x_ 0\| >u\}\quad as\quad u\to \infty, \] where \(x_ 0\in R^ n\), \(\mu\) is a symmetric Gaussian measure on \(R^ n\) and \(\| \cdot \|\) is a norm possessing some special properties. For this purpose Laplace’s method is extended to the integral \(\int_{K}e^{-u^ 2h(x)+ug(x)}dx\) for some functions h and g on a compact set \(K\subset R^ n\). The general results then are applied to \(l_ p\)-norms, \(p>0\).
Reviewer: R.Norvaiša (Vilnius)
