In this paper we extend results on reconstruction of probabilistic supports of independent and identically distributed random variables to supports of dependent stationary ${\mathbb R}^d$-valued random variables. All supports are assumed to be compact of positive reach in Euclidean space. Our main results involve the study of the convergence in the Hausdorff sense of a cloud of stationary dependent random vectors to their common support. A novel topological reconstruction result is stated, and a number of illustrative examples are presented. The example of the Möbius Markov chain on the circle is treated at the end with simulations.

This paper deals with the problem of estimating the level setsL(c) =  {F(x) ≥ c},with c ∈ (0,1), of an unknown distribution function F on ℝ+2. A plug-inapproach is followed. That is, given a consistent estimatorFn of F, we estimateL(c) byLn(c) =  {Fn(x) ≥ c}.In our setting, non-compactness property is a priori required for thelevel sets to estimate. We state consistency results with respect to the Hausdorffdistance and the volume of the symmetric difference. Our results are motivated byapplications in multivariate risk theory. In particular we propose a new bivariate versionof the conditional tail expectation by conditioning the two-dimensional random vector tobe in the level set L(c). We also present simulated andreal examples which illustrate our theoretical results.

This paper presents a novel method for localization of mobile robots in structured environments. The estimation of the position and orientation of the robot relies on the minimisation of the partial Hausdorff distance between ladar range measurements and a floor plan image of the building. The approach is employed in combination with an extended Kalman filter to obtain accurate estimates of the robot's position, heading and velocity. Good estimates of these variables were obtained during tests performed using a differential drive robot, thus demonstrating that the approach provides an accurate, reliable and computationally feasible alternative for indoor robot localization and autonomous navigation.

Let LSC(X) denote the set of extended real valued lower semicontinuous functions on a metrizable space X. If f, f1, f2, f3,... is a sequence in LSC(X), we say 〈fn〉 is epi-convergent to f provided the sequence of epigraphs 〈epi fn〉 is Kuratowski- Painlevé convergent to epi f. In this note we address the following question: what conditions on f and/or on X are necessary and sufficient for this mode of convergence to force epigraphical convergence with respect to the stronger Hausdorff metric and Vietoris topologies?

The chord length distributions of planar convex sets are discussed, particularly the density values at the extremes of the range; there is a qualitative distinction between polygons and sets with smooth boundaries. The distance between convex sets is related to the distance between distribution functions.