References

CHAINS OF DISTRIBUTIONS, HIERARCHICAL BAYESIAN MODELS AND BENFORD'S LAW


[1] F. Benford, The law of anomalous numbers, Proceedings of the American Philosophical Society 78 (1938), 551-572.

[2] A. Berger, Leonid A. Bunimovich and T. Hill, One-dimensional dynamical systems and Benford’s law, Trans. Amer. Math. Soc. 357(1) (2005), 197-219.

[3] A. Berger and T. Hill, Newton’s method obeys Benford’s law, The Amer. Math. Monthly 114(7) (2007), 588-601.

[4] J. O. Berger, Statistical Decision Theory and Bayesian Analysis, 2nd edition, Springer Series in Statistics, 1985.

[5] J. Brown and R. Duncan, Modulo one uniform distribution of the sequence of logarithms of certain recursive sequences, Fibonacci Quarterly 8 (1970), 482-486.

[6] E. Costas, V. López-Rodas, F. J. Toro and A. Flores-Moya, The number of cells in colonies of the cyanobacterium Microcystis aeruginosa satisfies BenfordŠs law, Acquatic Botany 89 (2008), DOI 10.1016/j.aquabot.2008.03.011.

[7] P. Diaconis, The distribution of leading digits and uniform distribution mod 1, Ann. Probab. 5 (1979), 72-81.

[8] P. Diaconis and L. Saloff-Coste, Comparison theorems for reversible Markov chains, Ann. Appl. Probab. 3(3) (1993), 696-730.

[9] R. W. Hamming, On the distribution of numbers, Bell Syst. Tech. J. 49 (1970), 1609-1625.

[10] W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57(1) (1970), 97-109.

[11] T. Hill, A statistical derivation of the significant-digit law, Statistical Science 10 (1996), 354-363.

[12] T. Hill, The first-digit phenomenon, American Scientists 86 (1996), 358-363.

[13] W. Hurlimann, Benford’s law from 1881 to 2006: a bibliography,
arxiv.org/abs/math/0607168

[14] D. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Addison-Wesley, 3rd edition, 1997.

[15] A. Kontorovich and S. J. Miller, Benford’s law, values of L-functions and the 3x+1 problem, Acta Arith. 120 (2005), 269-297.

[16] A. E. Kossovsky, Towards a better understanding of the leading digits phenomena, preprint.
arxiv.org/abs/math/0612627

[17] J. Lagarias and K. Soundararajan, Benford’s law for the 3x + 1 function, J. London Math. Soc. (2) 74(2) (2006), 289-303.

[18] Markov Chains and Mixing Times

[19] E. Ley, On the peculiar distribution of the U.S. stock indices digits, The American Statistician 50(4) (1996), 311-313.

[20] N. Madras and D. Randall, Markov chain decomposition for convergence rate analysis, Ann. Appl. Probab. 12(2) (2002), 581-606.

[21] W. Mebane, Election Forensics: The Second-digit Benford’s Law Test and Recent American Presidential Elections, Election Fraud Conference, Salt Lake City, Utah, September 29-30, 2006.
www.umich.edu/»wmebane/fraud06.pdf

[22] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, Journal of Chemical Physics 21(6) (1953), 1087-1092.

[23] S. J. Miller and R. Takloo-Bighash, An Invitation to Modern Number Theory, Princeton University Press, Princeton, NJ, 2006.

[24] S. J. Miller and M. Nigrini, The modulo 1 central limit theorem and Benford’s law for products, International Journal of Algebra 2(3) (2008), 119-130.

[25] S. Newcomb, Note on the frequency of use of the different digits in natural numbers, Amer. J. Math. 4 (1881), 39-40.

[26] M. Nigrini, Digital Analysis and the Reduction of Auditor Litigation Risk, Pages 69-81 in Proceedings of the 1996 Deloitte & Touche / University of Kansas Symposium on Auditing Problems, ed. M. Ettredge, University of Kansas, Lawrence, KS, 1996.

[27] M. Nigrini, The use of Benford’s law as an aid in analytical procedures, Auditing: A Journal of Practice & Theory 16(2) (1997), 52-67.

[28] M. Nigrini and S. J. Miller, Benford’s law applied to hydrology data - results and relevance to other geophysical data, Mathematical Geology 39(5) (2007), 469-490.

[29] Jean-Christophe Pain, Benford’s law and complex atomic spectra, Phys. Rev. E 77, 012102 (2008).

[30] S. J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Cambridge Studies in Advanced Mathematics 14, Cambridge University Press, 1995.

[31] R. A. Raimi, The first digit problem, Amer. Math. Monthly 83(7) (1976), 521-538.

[32] D. Randall, Rapidly mixing Markov chains with applications in computer science and physics, Computing in Science and Engineering 8(2) (March 2006), 30-41.

[33] A. J. Sinclair, Algorithms for Random Generation and Counting, A Markov Chain Approach. Progress in Theoretical Computer Science, Birkhäuser, Boston, MA, 1993.

[34] A. J. Sinclair, Convergence Rates for Monte Carlo Experiments. In Numerical Methods for Polymeric Systems (Minneapolis, MN, 1996), 1-17, S. G. Whittington, ed., IMA Vol. Math. Appl. 102, Springer, New York, 1998.

[35] E. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, Princeton, NJ, 2003.