Claude Bélisle

Hit-and-run algorithms for generating multivariate distributions

We introduce a general class of Hit-and-Run algorithms for generating essentially arbitrary absolutely continuous distributions on Rd. They include the Hypersphere Directions algorithm and the Coordinate Directions algorithm that have been proposed for identifying nonredundant linear constraints and for generating uniform distributions over subsets of Rd. Given a bounded open set S in Rd, an absolutely continuous probability distribution π on S the target distribution and an arbitrary probability distribution I? on the boundary of the d-dimensional unit sphere centered at the origin the direction distribution, the I?, π-Hit-and-Run algorithm produces a sequence of iteration points as follows. Given the nth iteration point x, choose a direction I? according to the distribution I? and then choose the n + 1st iteration point according to the conditionalization of the distribution π along the line defined by x and x + I?. Under mild conditions, we show that this sequence of points is a Harris recurrent reversible Markov chain converging in total variation to the target distribution π.

Convergence properties of hit–and–run samplers

Hit–and–Run algorithms are probabilistic methods for generating points at random according to some prescribed distribution π on a subset A of R d . Given a current point, say , a direction vector, say , is chosen at random according to some prescribed random mechanism. The next point , is then chosen at random according to the conditionalization of π on the line defined by Xk and . Under appropriate conditions, the sequence will be a Markov chain converging in total variation to the target distribution π. This paper introduces a new class of Hit–and–Run algorithms. A general convergence theorem is obtained and the existence, within this class, of particular Hit–and–Run algorithms with desirable asymptotic properties is established

Odd central moments of unimodal distributions

We present a simple geometric condition under which all existing odd central moments of a unimodal distribution are non-negative. The criterion applies to both the absolutely continuous case and the lattice case. In the lattice case, the result proves and generalizes a conjecture of Frame, Gilliland and Hsing. In the absolutely continuous case, the result provides a new proof of results of Hannan and Pitman, Runnenburg, and MacGillivray. The main idea is a new decomposition result for unimodal distributions.

Windings of spherically symmetric random walks via Brownian embedding

Let X1, X2, X3,... be a sequence of i.i.d. 2-valued random variables with a spherically symmetric distribution. Let (Sn; n[greater-or-equal, slanted]0) be its sequence of partial sums and let ([phi](n); n[greater-or-equal, slanted]0) be its winding sequence. Assuming only a mild moment condit show, via Brownian embedding, that 2[phi](n)/log n converges in distribution to a standard hyperbolic secant distribution.

Slow hit-and-run sampling ☆

We show that the hit-and-run sampler can converge to its target distribution at an arbitrarily slow rate. We also illustrate how the speed of convergence of the hit-and-run sampler can be affected by small perturbations of the target distribution.