Asymptotic uniformity of the quantization error of self-similar measures

Let??? be a self-similar measure on ${\mathbb{R}^d}$ associated with a family of contractive similitudes {S ₁, . . . , S _N } and a probability vector {p ₁, . . . , p _N }. Let ${(\alpha_n)_{n=1}^\infty}$ be a sequence of n-optimal sets for??? of order r. For each n, we denote by ${\{P_a(\alpha_n) : a \in \alpha_n\}}$ a Voronoi partition of ${\mathbb{R}^d}$ with respect to ?? _n . Under the strong separation condition for {S ₁, . . . , S _N }, we show that the nth quantization error of??? of order ${r \in [1, \infty)}$ satisfies the following asymptotic uniformity property: $$\int \limits _{P_a(\alpha_n)}{\rm d}(x, a)^rd\mu(x) \asymp \frac{1}{n}V_{n,r}(\mu),\quad {\rm for\,all}\,a \in \alpha_n.$$      

Asymptotic quantization of probability distributions

We give a brief introduction to results on the asymptotics of quantizatlon errors. The topics discussed in-clude the quantization dimension, asymptotic distributions of sets of prototypes, asymptotically optimalquantizations, approximations and random quantizations.     

A note on the quantization for probability measures with respect to the geometric mean error

We study the quantization with respect to the geometric mean error for probability measures μ on for which there exist some constants C, η > 0 such that for all ε > 0 and all . For such measures μ, we prove that the upper quantization dimension of μ is bounded from above by its upper packing dimension and the lower one is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.     

A note on the quantization for probability measures with respect to the geometric mean error

We study the quantization with respect to the geometric mean error for probability measures μ on \({\mathbb{R}^d}\) for which there exist some constants C, η > 0 such that \({\mu(B(x,\varepsilon))\leq C\varepsilon^\eta}\) for all ε > 0 and all \({x\in\mathbb{R}^d}\) . For such measures μ, we prove that the upper quantization dimension \({\overline{D}(\mu)}\) of μ is bounded from above by its upper packing dimension and the lower one \({\underline{D}(\mu)}\) is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.     

On the singularity of functions and the quantization of probability measures

Mathematical Notes -     

Some remarks on absolute continuity and quantization of probability measures

We introduce a notion of monotonicity of dimensions of measures. We show that the upper and lower quantization dimensions are not monotone. We give sufficient conditions in terms of so-called vanishing rates such that νμ implies . As an application, we determine the quantization dimension of a class of measures which are absolutely continuous w.r.t. some self-similar measure, with the corresponding Radon–Nikodym derivative bounded or unbounded. We study the set of quantization dimensions of measures which are absolutely continuous w.r.t. a given probability measure μ. We prove that the infimum on this set coincides with the lower packing dimension of μ. Furthermore, this infimum can be attained provided that the upper and lower packing dimensions of μ are equal.     

On the upper and lower quantization coefficient for probability measures on multiscale Moran sets

Given a finite set of patterns, we consider the Moran sets determined by using each of these patterns with a prescribed frequency. For certain infinite product measures μ on such Moran sets, we determine the exact values of the quantization dimensions D_r(μ). We give various sufficient conditions for the D_r(μ)-dimensional upper quantization coefficient and the lower one to be positive and finite. We also construct an example to illustrate our main result.     

Lattice quantization error for redundant representations

Redundant systems such as frames are often used to represent a signal for error correction, denoising and general robustness. In the digital domain quantization needs to be performed. Given the redundancy, the distribution of quantization errors can be rather complex. In this paper we study quantization error for a signal X in represented by a frame using a lattice quantizer. We completely characterize the asymptotic distribution of the quantization error as the cell size of the lattice goes to zero. We apply these results to get the necessary and sufficient conditions for the asymptotic form of the White Noise Hypothesis in the case of the pulse-code modulation scheme.     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic behavior of the threshold minimizing the average probability of error in calculation of wavelet coefficients

The problem of estimating the signal function from noisy observations by thresholding the coefficients of its wavelet decomposition is considered. The asymptotic orders of the threshold and risk are calculated by minimizing the average probability of error in calculating the wavelet coefficients.     

Asymptotic property for some series of probability

Let {X, X n , n≥1} be a sequence of i.i.d.random variables with zero mean, and set Sn = Σ k=1 n X k , EX2=σ 2>0, λ(ε) =Σ n=1 ∞ P (|Sn|≥ nε). In this paper, we discuss the rate of the approximation of σ2 by ε2 λ(ε) under suitable conditions, and improve the corresponding results of Klesov (1994).     

Asymptotic error expansions for hypersingular integrals

This paper presents quadrature formulae for hypersingular integrals $\int_a^b\frac{g(x)}{|x-t|^{1+\alpha }}\mathrm{d}x$ , where a?<?t?<?b and 0?<?α?≤?1. The asymptotic error estimates obtained by Euler–Maclaurin expansions show that, if g(x) is 2m times differentiable on [a,b], the order of convergence is O(h ^2μ ) for α?=?1 and O(h ^2μ???α ) for 0?<?α?<?1, where μ is a positive integer determined by the integrand. The advantages of these formulae are as follows: (1) using the formulae to evaluate hypersingular integrals is straightforward without need of calculating any weight; (2) the quadratures only involve g(x), but not its derivatives, which implies these formulae can be easily applied for solving corresponding hypersingular boundary integral equations in that g(x) is unknown; (3) more precise quadratures can be obtained by the Richardson extrapolation. Numerical experiments in this paper verify the theoretical analysis.     

Asymptotic formula for the error in cardinal interpolation

Summary. We give the asymptotic formula for the error in cardinal interpolation. We generalize the Mazur Orlicz Theorem for periodic function. Received February 22, 1999 / Revised version received October 15, 1999 / Published online March 20, 2001     

Asymptotic behavior of the prediction error

Let {X_n} _– be a random process, stationary in the broad sense, with spectral density f() satisfying the singularity condition: · We denote _n ² the mean square prediction error at the prediction of _o by linear forms in X_–1, ... , X_–n. In the paper one investigates the rate of decrease of _n to zero.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 11–24, 1983.In conclusion, the author wishes to express his gratitude to I. A. Ibragimov for his constant interest and help.     

Asymptotic expansion for the probability of large deviations. II

Lithuanian Mathematical Journal -