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\noindent Statistica Sinica {\bf 16}(2006), 77-91

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\centerline{\large\bf MEETING HAUSDORFF IN MONTE CARLO: }
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\centerline{\large\bf  A SURPRISING TOUR WITH ANTIHYPE FRACTALS}
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\author{Radu V. Craiu and Xiao-Li Meng}
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\centerline{\it University of Toronto and Harvard University}
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\begin{quotation}
\noindent {\it Abstract:}
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
{\it 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.\par
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\noindent {\it Key words and phrases:}
Antithetic variates, extreme
antithesis, fractals, Hausdorff dimension, Koch's curve, Latin
hypercube sampling, Markov chain Monte Carlo, self-similar
fractals.
\par
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