## Foundations of quantization for probability distributions.(English)Zbl 0951.60003

Lecture Notes in Mathematics. 1730. Berlin: Springer. x, 230 p. (2000).
The monograph provides the first systematic mathematical treatment of the following quantization problem: let $$\|\cdot\|$$ be a norm on $$\mathbb{R}^d$$, $$P$$ a probability measure on $$\mathbb{R}^d$$, $$n\in\mathbb{N}$$, and $$r\geq 1$$. Then $V_{n,r}(P): =\inf\Bigl\{\int \min_{a\in S}\|x-a\|^r dP(x):S\subset \mathbb{R}^d,\;|S|\leq n\Bigr\}$ is called the quantization error of $$P$$. It measures how well $$P$$ can be approximated by a probability measure which is supported by $$n$$ points. The problem of computing $$V_{n,r}(P)$$ and determining a minimizing set $$S$$ is of great importance, e.g. in engineering (signal compression) and operations research (optimal location of service centers). Since precise values of $$V_{n,r}(P)$$ and exact minimizers $$S$$ are known only in a few special cases, estimates and asymptotic studies for large $$n$$ are relevant.
In the first chapter the authors provide general properties of quantization, for example sufficient conditions for existence and uniqueness of minimizing sets $$S$$. Chapter 2 treats asymptotics in case $$P$$ is nonsingular w.r.t. Lebesgue measure. Theorem 6.2 states that under a slight moment condition, $$n^{r/d} V_{n,r}(P)$$ converges as $$n\to\infty$$ and a formula of the limit is provided in terms of a norm of the density of the absolutely continuous part $$P_a$$ of $$P$$. For any sequence $$S_n$$ of asymptotically minimizing sets it is shown in Theorem 7.5 that the associated empirical measure converges weakly to some measure $$P_r$$, whose density is given in terms of the density of $$P_a$$. There is also a section on random quantizers in which $$S=\{Y_1, \dots,Y_n\}$$, where $$Y_1,\dots, Y_n$$ are i.i.d. The third chapter contains a number of recent results by the authors on the asymptotics for singular probability measures on $$\mathbb{R}^d$$.
The lecture note is a well-written, detailed and comprehensive exposition of the state-of-the-art of the quantization problem for probability measures on $$\mathbb{R}^d$$.

### MSC:

 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 60E05 Probability distributions: general theory

### Keywords:

quantization; quantization error; singular measure
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