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 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.
Key words and phrases: Antithetic variates, extreme antithesis, fractals, Hausdorff dimension, Koch's curve, Latin hypercube sampling, Markov chain Monte Carlo, self-similar fractals.
<