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MEETING HAUSDORFF IN MONTE CARLO: A SURPRISING TOUR WITH ANTIHYPE FRACTALS
Radu V. Craiu and Xiao-Li Meng
Vol. 16, No. 1 (January 2006), pp. 77-91
Published by: Institute of Statistical Science, Academia Sinica
Stable URL: http://www.jstor.org/stable/24307480
Page Count: 15
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To many statistical researchers, fractals are aesthetically pleasing mathematical objects or ingredients of complex theoretical studies. This article documents an exception: during recent research on improving effectiveness of Markov chain Monte Carlo (MCMC), we unexpectedly encountered a class of intriguing fractals in the simple context of generating negatively correlated random variates that achieve extreme antithesis. This class of antihype fractals enticed us to tour the world of fractals, because it has intrinsic connections with classical fractals such as Koch's snowflake and it illustrates theoretical concepts such as Hausdorff dimension in a very intuitive way. It also provides a practical example where a sequence of uniform variables converges exponentially in the Kolmogorov-Smirnov distance, yet fails to converge in other common distances, including total variation distance and Hellinger distance. We also show that this non-convergence result actually holds for any sequence of (proper) uniform distributions on supports formed by the generating process of a self-similar fractal. These negative results remind us that the choice of metrics, e.g., for diagnosing convergence of MCMC algorithms, do matter sometimes in practice.
Statistica Sinica © 2006 Institute of Statistical Science, Academia Sinica