×

Moments of distributions related to digital expansions. (English) Zbl 1086.60009

Summary: The paper studies certain (probability) measures of binomial type defined in a recursive way on the unit interval. These measures are related to the sum-of-digit function and similar quantities. In particular, we undertake an asymptotic analysis of the moments of the corresponding distributions. This is done by a combination of a method based on the Mellin transform and the depoissonisation technique.

MSC:

60E05 Probability distributions: general theory
60E10 Characteristic functions; other transforms
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bassino, F.; Prodinger, H., \((q,δ)\)-numeration systems with missing digits, Monatsh. Math., 141, 89-99 (2004) · Zbl 1055.11047
[2] David, G.; Semmes, S., Fractured fractals and broken dreams: Self-similar geometry through metric and measure, (Oxford Lecture Ser. Math. Appl., vol. 7 (1997), Clarendon, Oxford Univ. Press: Clarendon, Oxford Univ. Press New York) · Zbl 0887.54001
[3] Delange, H., Sur la fonction sommatoire de la fonction “somme des chiffres,”, Enseign. Math. (2), 21, 31-47 (1975) · Zbl 0306.10005
[4] Falconer, K. J., Fractal Geometry. Mathematical Foundations and Applications (1990), Wiley: Wiley Chichester · Zbl 0689.28003
[5] Fischer, H.-J., On the paper: Asymptotics for the moments of singular distributions, J. Approx. Theory, 82, 362-374 (1995), [J. Approx. Theory 74 (1993) 301-334; MR 94h:44002] by W. Goh and J. Wimp · Zbl 0830.41025
[6] Flajolet, Ph.; Gourdon, X.; Dumas, Ph., Mellin transforms and asymptotics: Harmonic sums, Theoret. Comput. Sci., 144, 3-58 (1995) · Zbl 0869.68057
[7] Flajolet, Ph.; Grabner, P.; Kirschenhofer, P.; Prodinger, H.; Tichy, R. F., Mellin transforms and asymptotics: Digital sums, Theoret. Comput. Sci., 123, 291-314 (1994) · Zbl 0788.44004
[8] Goh, W.; Wimp, J., Asymptotics for the moments of singular distributions, J. Approx. Theory, 74, 301-334 (1993) · Zbl 0788.41019
[9] Goh, W.; Wimp, J., A generalized Cantor-Riesz-Nagy function the growth of its moments, Asymptot. Anal., 8, 379-392 (1994) · Zbl 0810.26003
[10] Grabner, P. J.; Prodinger, H., Asymptotic analysis of the moments of the Cantor distribution, Statist. Probab. Lett., 26, 243-248 (1996) · Zbl 0844.62016
[11] Jacquet, Ph.; Szpankowski, W., Analytical de-Poissonization and its applications, Theoret. Comput. Sci., 201, 1-62 (1998) · Zbl 0902.68087
[12] Knuth, D., The Art of Computer Programming, vol. 4, Pre-Fascicle 2a: Generating all \(n\)-tuples (version of 29 Aug. 2003) (2004)
[13] Kobayashi, Z., Digital sum problems for the Gray code representation of natural numbers, Interdiscip. Inform. Sci., 8, 167-175 (2002) · Zbl 1018.11005
[14] Muramoto, K.; Okada, T.; Sekiguchi, T.; Shiota, Y., Digital sum problems for the \(p\)-adic expansion of natural numbers, Interdiscip. Inform. Sci., 6, 105-109 (2000) · Zbl 0987.11006
[15] Muramoto, K.; Okada, T.; Sekiguchi, T.; Shiota, Y., An explicit formula of subblock occurrences for the \(p\)-adic expansion, Interdiscip. Inform. Sci., 8, 115-121 (2002) · Zbl 1014.11008
[16] Okada, T.; Sekiguchi, T.; Shiota, Y., Applications of binomial measures to power sums of digital sums, J. Number Theory, 52, 256-266 (1995) · Zbl 0824.11004
[17] Okada, T.; Sekiguchi, T.; Shiota, Y., An explicit formula of the exponential sums of digital sums, Japan J. Indust. Appl. Math., 12, 425-438 (1995) · Zbl 0844.11007
[18] Okada, T.; Sekiguchi, T.; Shiota, Y., A generalization of Hata-Yamaguti’s results on the Takagi function II: Multinomial case, Japan J. Indust. Appl. Math., 13, 435-463 (1996) · Zbl 0871.26006
[19] Shiota, Y.; Sekiguchi, T.; Okada, T., Hata-Yamaguti’s result on [the] Takagi function and its applications to digital sum problems, Sūrikaisekikenkyūsho Kōkyūroku, 961, 73-80 (1996), Analytic number theory, Kyoto, 1995 (in Japanese) · Zbl 1043.11540
[20] Szpankowski, W., Average Case Analysis of Algorithms on Sequences, Wiley-Intersci. Ser. Discrete Math. Optim. (2001), Wiley-Interscience: Wiley-Interscience New York, with a foreword by Philippe Flajolet · Zbl 0969.00028
[21] Weisstein, E., Mathworld
[22] Woess, W., Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Math., vol. 138 (2000), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0951.60002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.