Asymptotic properties of digit sequences of random numbers

Autor(en): Stadje, W 
Stichwörter: ACKNOWLEDGMENT; ergodicity; limit theorem; Markov chain; Mathematics; moments; occurrence probability; PRIORITY; random digit sequence; RANDOM-VARIABLES; SERIES; stationary digits; transition function
Erscheinungsdatum: 2005
Herausgeber: WILEY-V C H VERLAG GMBH
Journal: MATHEMATISCHE NACHRICHTEN
Volumen: 278
Ausgabe: 10
Startseite: 1209
Seitenende: 1229
Zusammenfassung: 
We consider several aspects of the relationship between a [0, 1]-valued random variable X and the random sequence of digits given by its m-ary expansion. We present results for three cases: (a) independent and identically distributed digit sequences; (b) random variables X with smooth densities; (c) stationary digit sequences. In the case of i.i.d. an integral limit thorem is proved which applies for example to relative frequencies, yielding asymptotic moment identities. We deal with occurrence probabilities of digit groups in the case that X has an analytic Lebesgue density. In the case of stationary digits we determine the distribution of X in terms of their transition functions. We study an associated [0, 1]-valued Markov chain, in particular its ergodicity, and give conditions for the existence of stationary digit sequences with prespecified transition functions. It is shown that all probability measures induced on [0, 1] by such sequences are purely singular except for the uniform distribution. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
ISSN: 0025584X
DOI: 10.1002/mana.200310303

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