Continued fractions built from convex sets and convex functions

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions with deterministic terms are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform (where also necessary and sufficient conditions of convergence for continued fractions with constant terms are obtained) and the family of non-negative convex functions with the Legendre--Fenchel and Artstein-Avidan--Milman transforms.Comment: 18 pages. This version deals with the deterministic case only and is due to appear in Communications in Contemporary Mathematics. The random case will be posted separatel

Level sets estimation and Vorob'ev expectation of random compact sets

The issue of a "mean shape" of a random set $X$ often arises, in particular in image analysis and pattern detection. There is no canonical definition but one possible approach is the so-called Vorob'ev expectation \E_V(X), which is closely linked to quantile sets. In this paper, we propose a consistent and ready to use estimator of \E_V(X) built from independent copies of $X$ with spatial discretization. The control of discretization errors is handled with a mild regularity assumption on the boundary of $X$: a not too large 'box counting' dimension. Some examples are developed and an application to cosmological data is presented

On the domain of attraction for the lower tail in Wicksell's corpuscle problem

We consider the classical Wicksell corpuscle problem with spherical particles in R^n and investigate the shapes of lower tails of distributions of `sphere radii' in R^n and `sphere radii' in a k-dimensional section plane. We show in which way the domains of attraction are related to each other.Comment: 6 page

Band depths based on multiple time instances

Bands of vector-valued functions $f:T\mapsto\mathbb{R}^d$ are defined by considering convex hulls generated by their values concatenated at $m$ different values of the argument. The obtained $m$-bands are families of functions, ranging from the conventional band in case the time points are individually considered (for $m=1$) to the convex hull in the functional space if the number $m$ of simultaneously considered time points becomes large enough to fill the whole time domain. These bands give rise to a depth concept that is new both for real-valued and vector-valued functions.Comment: 12 page

Multifractional Poisson process, multistable subordinator and related limit theorems

We introduce a multistable subordinator, which generalizes the stable subordinator to the case of time-varying stability index. This enables us to define a multifractional Poisson process. We study properties of these processes and establish the convergence of a continuous-time random walk to the multifractional Poisson process.Comment: Revisio