高斯随机场 $L_p$-范数的上函数
概率论
2013-11-21 v1
摘要
本文旨在寻找随机变量集合 { ∥ ξ h ⃗ ∥ p , h ⃗ ∈ H } , 1 ≤ p < ∞ \big\{\big\|\xi_{\vec{h}}\big\|_p, \vec{h}\in\mathrm{H}\big\}, 1\leq p<\infty { ξ h p , h ∈ H } , 1 ≤ p < ∞ ξ h ⃗ ( x ) , x ∈ ( − b , b ) d , d ≥ 1 \xi_{\vec{h}}(x), x\in(-b,b)^d, d\geq 1 ξ h ( x ) , x ∈ ( − b , b ) d , d ≥ 1 ∥ ⋅ ∥ p \|\cdot\|_p ∥ ⋅ ∥ p ( − b , b ) d (-b,b)^d ( − b , b ) d L p L_p L p H \mathrm{H} H ( − b , b ) d (-b,b)^d ( − b , b ) d R + d R^d_+ R + d d d d { Ψ α ( h ⃗ ) , h ⃗ ∈ H } \left\{\Psi_\alpha\big(\vec{h}\big),\;\;\vec{h}\in\mathrm{H}\right\} { Ψ α ( h ) , h ∈ H } E { sup h ⃗ ∈ H [ ∥ ξ h ⃗ ∥ p − Ψ α ( h ⃗ ) ] + } q ≤ α q , q ≥ 1 , E\big\{\sup_{\vec{h}\in\mathrm{H}}\big[\big\|\xi_{\vec{h}}\big\|_p-\Psi_\alpha\big(\vec{h}\big)\big]_+\big\}^q\leq \alpha^q,\; q\geq 1, E { sup h ∈ H [ ξ h p − Ψ α ( h ) ] + } q ≤ α q , q ≥ 1 , α > 0 \alpha>0 α > 0
引用
@article{arxiv.1311.4996,
title = {Upper functions for $L_p$-norm of gaussian random fields},
author = {O. Lepski},
journal= {arXiv preprint arXiv:1311.4996},
year = {2013}
}