Functional quantization of Gaussian processes
Functional quantization of Gaussian processes
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DOI：
10.1016/s0022-1236(02)00010-1
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发表时间：
2002-12-20
影响因子：
1.7
通讯作者：
Pagès, G
中科院分区：
文献类型：
--
作者：
Luschgy, H;Pagès, G
量化在于研究由采用有限数量 n 个值的向量（量化版本）近似随机向量 X 引起的 L-r 误差。对于 R-m 值随机向量，理论和实践已经相当成熟，特别是，非奇异分布所产生的最小量化误差 n --> 无穷大的渐进性是众所周知的：它的行为类似于 c(X,r,m)n(-1/m)。本文将该问题转置为无限维希尔伯特空间中的随机向量，特别是随机过程 (X-t)（是 [0,1] 的元素），将其视为 L-2([0, 1],dt) 值随机向量。对于高斯向量和 L-2 误差，我们提供了固定量化器和最优量化器的详细结果。我们进一步在速率问题和 X 的香农-柯尔莫哥洛夫熵之间建立了精确的联系。这使我们能够在协方差算子的特征值的相当一般的条件下计算最小 L-2 量化误差收敛到零的精确速率。典型的速率为 O((log n)(-a))，a > 0。例如，它们是针对分数布朗运动和分数 Ornstein-Uhlenbeck 过程获得的。指数a与过程的L-2正则性密切相关。 (C) 2002 年爱思唯尔科学（美国）。版权所有。
Quantization consists in studying the L-r-error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. For R-m-valued random vectors the theory and practice is quite well established and in particular, the asymptotics as n --> infinity of the resulting minimal quantization error for nonsingular distributions is well known: it behaves like c(X,r,m)n(-1/m). This paper is a transposition of this problem to random vectors in an infinite dimensional Hilbert space and in particular, to stochastic processes (X-t)(tis an element of[0,1]) viewed as L-2([0, 1],dt)-valued random vectors. For Gaussian vectors and the L-2-error we present detailed results for stationary and optimal quantizers. We further establish a precise link between the rate problem and Shannon-Kolmogorov's entropy of X. This allows us to compute the exact rate of convergence to zero of the minimal L-2-quantization error under rather general conditions on the eigenvalues of the covariance operator. Typical rates are O((log n)(-a)), a > 0. They are obtained, for instance, for the fractional Brownian motion and the fractional Ornstein-Uhlenbeck process. The exponent a is closely related with the L-2-regularity of the process. (C) 2002 Elsevier Science (USA). All rights reserved.