Previous |  Up |  Next


MCMC; Bernoulli convolution; binomial measure; Monte Carlo integration; empirical measures
We apply a Markov chain Monte Carlo method to approximate the integral of a continuous function with respect to the asymmetric Bernoulli convolution and, in particular, with respect to a binomial measure. This method---inspired by a cognitive model of memory decay---is extremely easy to implement, because it samples only Bernoulli random variables and combines them in a simple way so as to obtain a sequence of empirical measures converging almost surely to the Bernoulli convolution. We give explicit bounds for the bias and the standard deviation for this approximation, and present numerical simulations showing that it outperforms a general Monte Carlo method using the same number of Bernoulli random samples.
[1] Andrieu, C., Freitas, N. De, Doucet, A., Jordan, M. I.: An introduction to MCMC for machine learning. Mach. Learn. 50 (2003), 5-43. DOI 10.1023/A:1020281327116 | Zbl 1033.68081
[2] Barnsley, M. F., Demko, S.: Iterated function systems and the global construction of fractals. Proc. R. Soc. Lond., Ser. A 399 (1985), 243-275. DOI 10.1098/rspa.1985.0057 | MR 0799111 | Zbl 0588.28002
[3] Berkes, I., Csáki, E.: A universal result in almost sure central limit theory. Stoch. Proc. Appl. 94 (2001), 105-134. DOI 10.1016/S0304-4149(01)00078-3 | MR 1835848 | Zbl 1053.60022
[4] Calabrò, F., Esposito, A. Corbo: An efficient and reliable quadrature algorithm for integration with respect to binomial measures. BIT 48 (2008), 473-491 \MR 2447981. DOI 10.1007/s10543-008-0168-x | MR 2447981
[5] Dovgoshey, O., Martio, O., Ryazanov, V., Vuorinen, M.: The Cantor function. Expo. Math. 24 (2006), 1-37. DOI 10.1016/j.exmath.2005.05.002 | MR 2195181 | Zbl 1098.26006
[6] Jessen, B., Wintner, A.: Distribution functions and the Riemann zeta function. Trans. Am. Math. Soc. 38 (1935), 48-88. DOI 10.1090/S0002-9947-1935-1501802-5 | MR 1501802 | Zbl 0014.15401
[7] Kalos, M. H., Whitlock, P. A.: Monte Carlo Methods. Vol. I: Basics. Wiley New York (1986). MR 0864827
[8] Mandelbrot, B. B., Calvet, L., Fisher, A.: A multifractal model of asset returns. Cowles Foundation Discussion Papers: 1164 (1997), Retrieved from\_pdfs/Cowles1164.pdf (last access August 21, 2012).
[9] Peres, Y., Schlag, W., Solomyak, B.: Sixty years of Bernoulli convolutions. Fractal Geometry and Stochastics II. Proceedings of the 2nd conference Greifswald/Koserow, Germany, August 28--September 2, 1998. Eds. C. Bandt at al. Prog. Probab. 46 (2000), 39-65. MR 1785620
[10] Riedi, R. H.: Introduction to multifractals. Techn. Rep Rice Univ., October 26, 1999 Retrieved from (last access August 21, 2012).
[11] Strichartz, R. S., Taylor, A., Zhang, T.: Densities of self-similar measures on the line. Exp. Math. 4 (1995), 101-128. DOI 10.1080/10586458.1995.10504313 | MR 1377413 | Zbl 0860.28005
[12] White, K. G.: Forgetting functions. Animal Learning & Behavior 29 (2001), 193-207. DOI 10.3758/BF03192887
Partner of
EuDML logo