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Asymptotic properties of digit sequences of random numbers. (English) Zbl 1073.11053
Summary: 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.
##### MSC:
 11K31 Special sequences 60G10 Stationary stochastic processes 60J05 Discrete-time Markov processes on general state spaces 60F99 Limit theorems in probability theory
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##### References:
 [1] Barakat, Inform. Sci. 14 pp 189– (1978) [2] Blum, J. Math. Sci. 2 pp 1– (1967) [3] Boyle, Amer. Math. Monthly 101 pp 879– (1994) [4] Candelolo, Atti Sem. Mat. Fis. Univ. Modena 46 pp 511– (1998) [5] Chatterji, J. London Math. Soc. (2) 38 pp 375– (1963) [6] Chatterji, J. London Math. Soc. (2) 39 [7] Chatterji, Z. Wahrscheinlichkeitsth. verw. Geb. 3 pp 184– (1964) [8] Chatterji, Nederl. Akad. Wetensch. Proc. Ser. A 68 pp 754– (1965) [9] Hill, Amer. Math. Monthly 102 pp 322– (1995) [10] Hill, IEEE Trans. Inform. Theory IT-19 pp 326– (1973) [11] Kac, Bull. Amer. Math. Soc. (N. S.) 53 pp 1002– (1947) [12] Kasteleyn, J. Statist. Phys. 46 pp 811– (1987) [13] Marsaglia, Ann. Statist. 42 pp 1922– (1971) [14] Marsaglia, Ann. Probab. 2 pp 747– (1974) [15] Marsaglia, Ann. Statist. 2 pp 848– (1974) [16] McLaughlin, Fibonacci Quart. 22 pp 105– (1984) [17] Ravishankar, J. Statist. Phys. 53 pp 977– (1988) [18] Reich, Ann. Probab. 10 pp 787– (1982) [19] Robbins, Proc. Amer. Math. Soc. 4 pp 786– (1953) [20] , and , On the existence of the density of weighted sums, in: Limit Theorems in Probability Theory and Related Fields, Wissenschaft, Theorie und Praxis (TU Dresden, 1987), pp. 89-102. [21] Schatte, Statist. Probab. Lett. 37 pp 391– (1998) [22] Stadje, Math. Nachr. 121 pp 179– (1985) [23] , Average Case Analysis of Algorithms on Sequences (Wiley, New York, 2001). [24] , Coupling, Stationarity and Regeneration (Springer, New York, 2000). · Zbl 0949.60007
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