Probability Measures and Kolmogorov’s Theorem on Construction of Stochastic Processes

2019 ◽  
pp. 194-196
Keyword(s):  
Stochastic Processes ◽  
Probability Measures ◽  
Kolmogorov's Theorem

Fractal functional quantization of mean-regular stochastic processes

2010 ◽  
Vol 150 (1) ◽  
pp. 167-191 ◽  
Author(s):  
SIEGFRIED GRAF ◽  
HARALD LUSCHGY ◽  
GILLES PAGÈS
Keyword(s):  
Stochastic Processes ◽  
Probability Measures ◽  
Fractal Measure ◽  
Quantization Dimension ◽  
Regularity Index ◽  
Self Similar ◽  
The Mean ◽  
Functional Quantization ◽  
Borel Probability ◽  
Quantization Problem

AbstractWe investigate the functional quantization problem for stochastic processes with respect toLp(IRd, μ)-norms, where μ is a fractal measure namely, μ is self-similar or a homogeneous Cantor measure. The derived functional quantization upper rate bounds are universal depending only on the mean-regularity index of the process and the quantization dimension of μ and as universal rates they are optimal. Furthermore, for arbitrary Borel probability measures μ we establish a (nonconstructive) link between the quantization errors of μ and the functional quantization errors of the process in the spaceLp(IRd, μ).

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