Title | Functional quantization of Gaussian processes |

Publication Type | Journal Article |

Year of Publication | 2002 |

Authors | Harald Luschgy, and Gilles Pagès |

Journal | Journal of Functional Analysis |

Volume | 196 |

Pagination | 486–531 |

Date Published | December |

ISSN | 0022-1236 |

Keywords | fractional Brownian motion, Gaussian process, quantization of probability distribution, Shannon–Kolmogorov entropy, stationary processes |

Abstract | Quantization consists in studying the -error induced by the approximation of a random vector by a vector (quantized version) taking a finite number of values. For -valued random vectors the theory and practice is quite well established and in particular, the asymptotics as of the resulting minimal quantization error for nonsingular distributions is well known: it behaves like . This paper is a transposition of this problem to random vectors in an infinite dimensional Hilbert space and in particular, to stochastic processes viewed as -valued random vectors. For Gaussian vectors and the -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 : This allows us to compute the exact rate of convergence to zero of the minimal -quantization error under rather general conditions on the eigenvalues of the covariance operator. Typical rates are . They are obtained, for instance, for the fractional Brownian motion and the fractional Ornstein–Uhlenbeck process. The exponent a is closely related with the -regularity of the process. |