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Summary:In recent years optimal portfolio selection strategies for sequential investment have been shown to exist. Although their asymptotical optimality is well established, finite sample properties do need the adjustment of parameters that depend on dimensionality and scale. In this paper we introduce some nearest neighbor based portfolio selectors that solve these problems, and we show that they are also log-optimal for the very general class of stationary and ergodic random processes. The newly proposed algorithm shows very good finite-horizon performance when applied to different markets with different dimensionality or scales without any change: we see it as a very robust strategy.

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n independent random points drawn from a density f in R d define a random Voronoi partition. We study the measure of a typical cell of the partition. We prove that the asymptotic distribution of the probability measure of the cell centered at a point x ∈ R d is independent of x and the density f . We determine all moments of the asymptotic distribution and show that the distribution becomes more concentrated as d becomes large. In particular, we show that the variance converges to zero exponentially fast in d. We also obtain a density-free bound for the rate of convergence of the diameter of a typical Voronoi cell.

For a linear regression function the average of stochastic approximation with constant gain is considered. In case of ergodic observations almost sure convergence is proved, where the limit is biased with small bias for small gain. For independent and identically distributed observations and also under martingale and mixing assumptions, asymptotic normality with (n-1/2 )-convergence order is obtained. In the martingale case the asymptotic covariance matrix is close to the optimum one if the gain is small.