Simple Monte Carlo integration with respect to Bernoulli convolutions. (English) Zbl 1274.65003

Gómez, David M.; Dartnell, Pablo

Appl. Math., Praha 57, No. 6, 617-626 (2012).

Summary: 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.

MSC:

65C05	Monte Carlo methods
65D30	Numerical integration
60G57	Random measures

Keywords:

Bernoulli convolution; binomial measure; Monte Carlo integration; empirical measures

References:

[1]	C. Andrieu, N. De Freitas, A. Doucet, M. I. Jordan: An introduction to MCMC for machine learning. Mach. Learn. 50 (2003), 5–43. · Zbl 1033.68081 · doi:10.1023/A:1020281327116
[2]	M.F. Barnsley, S. Demko: Iterated function systems and the global construction of fractals. Proc. R. Soc. Lond., Ser. A 399 (1985), 243–275. · Zbl 0588.28002 · doi:10.1098/rspa.1985.0057
[3]	I. Berkes, E. Csáki: A universal result in almost sure central limit theory. Stoch. Proc. Appl. 94 (2001), 105–134. · Zbl 1053.60022 · doi:10.1016/S0304-4149(01)00078-3
[4]	F. Calabrò, A. Corbo Esposito: An efficient and reliable quadrature algorithm for integration with respect to binomial measures. BIT 48 (2008), 473–491. · Zbl 1155.65022 · doi:10.1007/s10543-008-0168-x
[5]	O. Dovgoshey, O. Martio, V. Ryazanov, M. Vuorinen: The Cantor function. Expo. Math. 24 (2006), 1–37. · Zbl 1098.26006 · doi:10.1016/j.exmath.2005.05.002
[6]	B. Jessen, A. Wintner: Distribution functions and the Riemann zeta function. Trans. Am. Math. Soc. 38 (1935), 48–88. · JFM 61.0462.03 · doi:10.1090/S0002-9947-1935-1501802-5
[7]	M.H. Kalos, P.A. Whitlock: Monte Carlo Methods. Vol. I: Basics. Wiley, New York, 1986. · Zbl 0655.65004
[8]	B.B. Mandelbrot, L. Calvet, A. Fisher: A multifractal model of asset returns. Cowles Foundation Discussion Papers: 1164. 1997, Retrieved from http://users.math.yale.edu/users/mandelbrot/web pdfs/Cowles1164.pdf (last access August 21, 2012).
[9]	Y. Peres, W. Schlag, B. Solomyak: 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.
[10]	R.H. Riedi: Introduction to multifractals. Techn. Rep. Rice Univ., October 26, 1999, Retrieved from http://rudolf.riedi.home.hefr.ch/Publ/PDF/intro.pdf (last access August 21, 2012).
[11]	R. S. Strichartz, A. Taylor, T. Zhang: Densities of self-similar measures on the line. Exp. Math. 4 (1995), 101–128. · Zbl 0860.28005 · doi:10.1080/10586458.1995.10504313
[12]	K.G. White: Forgetting functions. Animal Learning & Behavior 29 (2001), 193–207. · doi:10.3758/BF03192887