Asymptotic quantization errors for unbounded quantizers. (English) Zbl 1164.60377

Shykula, M.

Teor. Jmovirn. Mat. Stat. 75, 165-175 (2006).

The author considers the quantization of a random variable, which is the value of a random process at a fixed sampling point. Let \(X\) be a continuous random variable with density function \(f(x)\), \(x\in \mathbb R\). Let \(g(x)\) be a quantization density function and \(Q_{N,G}(X)\) the non-uniform companding quantizer. It is proved that if the quantization density function \(g(x)\) is positive and uniformly continuous and there exists \(C>0\) such that \(f(x)\leq Cg(x)\) for every \(x\in\mathbb R\), then (i) \(g(X)N(X-Q_{N,G}(X))\to U\) in distribution, as \(N\to\infty\); (ii)\(N(X-Q_{N,G}(X))\to U/g(X)\) in distribution, as \(N\to\infty\), where \(X\) and \(U\) are independent random variables; \(G(x)\) is a distribution function corresponding to \(g(x)\); \(U\) is a random variable uniformly distributed on \([-1/2,1/2]\).

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60G99	Stochastic processes
94A29	Source coding
94A34	Rate-distortion theory in information and communication theory

Keywords:

non-uniform scalar quantization; random process; stochastic structure