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 Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 2, Pages 386–395 (Mi tvp472)

Short Communications

Estimates for the Syracuse problem via a probabilistic model

K. A. Borovkova, D. Pfeiferb

a University of Melbourne, Department of Mathematics and Statistics
b Institut für Mathematische Stochastik, Universität, Germany

Abstract: We employ a simple stochastic model for the Syracuse problem (also known as the $(3x+ 1)$ problem) to get estimates for the average behavior of the trajectories of the original deterministic dynamical system. The use of the model is supported not only by certain similarities between the governing rules in the systems, but also by a qualitative estimate of the rate of approximation. From the model, we derive explicit formulae for the asymptotic densities of some sets of interest for the original sequence. We also approximate the asymptotic distributions for the stopping times (times until absorption in the only known cycle $\{1,2\}$) of the original system and give numerical illustrations of our results.

Keywords: Syracuse problem, dynamical system, random walk.

DOI: https://doi.org/10.4213/tvp472

Full text: PDF file (671 kB)

English version:
Theory of Probability and its Applications, 2001, 45:2, 300–310

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Citation: K. A. Borovkov, D. Pfeifer, “Estimates for the Syracuse problem via a probabilistic model”, Teor. Veroyatnost. i Primenen., 45:2 (2000), 386–395; Theory Probab. Appl., 45:2 (2001), 300–310

Citation in format AMSBIB
\Bibitem{BorPfe00} \by K.~A.~Borovkov, D.~Pfeifer \paper Estimates for the Syracuse problem via a~probabilistic model \jour Teor. Veroyatnost. i Primenen. \yr 2000 \vol 45 \issue 2 \pages 386--395 \mathnet{http://mi.mathnet.ru/tvp472} \crossref{https://doi.org/10.4213/tvp472} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1967765} \zmath{https://zbmath.org/?q=an:0984.60050} \transl \jour Theory Probab. Appl. \yr 2001 \vol 45 \issue 2 \pages 300--310 \crossref{https://doi.org/10.1137/S0040585X97978245} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000169004700009} 

• http://mi.mathnet.ru/eng/tvp472
• https://doi.org/10.4213/tvp472
• http://mi.mathnet.ru/eng/tvp/v45/i2/p386

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Sinai Y.G., “Statistical (3x+1) problem”, Communications on Pure and Applied Mathematics, 56:7 (2003), 1016–1028
2. Applegate D., Lagarias J.C., “Lower bounds for the total stopping time of 3x+1 iterates”, Mathematics of Computation, 72:242 (2003), 1035–1049
3. Lagarias J.C., Soundararajan K., “Benford's law for the 3 alpha+1 function”, Journal of the London Mathematical Society–Second Series, 74:2 (2006), 289–303
4. Doumas A.V., Papanicolaou V.G., “a Randomized Version of the Collatz 3X+1 Problem”, Stat. Probab. Lett., 109 (2016), 39–44
5. Rozier O., “The 3X”, Funct. Approx. Comment. Math., 56:1 (2017), 7–23
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