Fractal sums of pulses are defined on R D as follows:where the X n are independent random variables, H ∈]0, 1[ and G is the elementary "bump" or "pulse". If the X n are uniform on a cube and G sufficiently regular we prove the (almost sure) existence of F and show that the box dimension of its graph is 2 − H.

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We consider random functions formed as sums of pulses where Xn are independent random vectors, 0<α<1, and G is an elementary “pulse” or “bump”. Typically such functions have fractal graphs and we find the Hausdorff dimension of these graphs using a novel variant on the potential theoretic method.

Abstract. The modelling of random bi-phasic, or porous, media has been, and still is, under active investigation by mathematicians, physicists or physicians. In this paper we consider a thresholded random process X as a source of the two phases. The intervals when X is in a given phase, named chords, are the subject of interest. We focus on the study of the tails of the chord-length distribution functions. In the literature, different types of the tail behavior have been reported, among them exponential-like or power-like decay. We look for the link between the dependence structure of the underlying thresholded process X and the rate of decay of the chord-length distribution. When the process X is a stationary Gaussian process, we relate the latter to the rate at which the covariance function of X decays at large lags. We show that exponential, or nearly exponential, decay of the tail of the distribution of the chord-lengths is very common, perhaps surprisingly so.

Let K be a convex body in R 2 . We consider the Voronoi tessellation generated by a homogeneous Poisson point process of intensity λ conditional on the existence of a cell K λ which contains K. When λ → ∞, this cell K λ converges from above to K and we provide the precise asymptotics of the expectation of its defect area, defect perimeter and number of vertices. As in Rényi and Sulanke's seminal papers on random convex hulls, the