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Probabilities of randomly centered small balls and quantization in Banach spaces. (English) Zbl 1078.60029

A ball \(B\) with small radius \(r\) has a random centre \(X\) defined by a given probability law \(p\). The paper extends from Hilbert space to Banach space some properties of the random information density \(-\ln p(B(X,r))\). Some general properties of RSB (random small balls) are displayed, one proves the equivalence of RSB probabilities with nonrandom function under weak assumptions, one studies an alternative representation using gauge functions, one establishes the equivalence of small ball probabilities with random quantization, and lastly, it is shown that polynomial equivalents exist for RSB.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
46F25 Distributions on infinite-dimensional spaces
94A15 Information theory (general)
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