Functional Quantization and Small Ball Probabilities for Gaussian Processes
Functional Quantization and Small Ball Probabilities for Gaussian Processes
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高斯过程的函数量化和小球概率
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
H. Luschgy
中科院分区：
文献类型：
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作者：
S. Graf;H. Luschgy
量化就是研究一个随机向量X被一个取有限n个值的向量（量化版）逼近所引起的lr误差。我们研究了无限维巴拿赫空间中的高斯随机向量，特别是高斯过程的这一问题。Fehringer(4)和Dereich等人(3)证明了小球概率的下界和上界与量化误差的上界和下界之间的精确联系。我们通过证明从量化误差到小球概率的方向相同，建立了一个完整的关系。这允许我们从对数小球渐近中计算最小lr量化误差收敛到零的精确速率，反之亦然。
Quantization consists in studying the Lr-error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer(4) and Dereich et al.(3) relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal Lr-quantization error from logarithmic small ball asymptotics and vice versa.