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Asymptotic properties of digit sequences of random numbers
Author(s) -
Stadje Wolfgang
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310303
Subject(s) - mathematics , random variable , markov chain , independent and identically distributed random variables , ergodicity , limit (mathematics) , sequence (biology) , distribution (mathematics) , moment (physics) , lebesgue integration , stationary distribution , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , statistics , physics , classical mechanics , biology , genetics
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. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)