AN APPROACH OF MEAN-SETS THEORY FOR NEGATIVELY CURVED CONVEX COMBINATION POLISH METRIC SPACES

Abstract

The choice of a convenient approach to be used is one important issue when attempting to develop methods or obtain results in the setting of Probability on general metric spaces. In this paper, we extend the mean-sets probabilistic approach formally introduced by Mosina on locally finite graphs, and hence (via Cayley graphs) on finitely generated groups, to the field of Negatively Curved Convex Combination Polish (NCCCP) metric spaces. We construct an appropriated Vertex-Weighted Metric (VWM) graph in the framework of this class of geometrical structures. We define a function called convexification function on the direct product of $ n $ copies of the vertex-set of this graph (for a given fixed integer $ n\geq 2$), using the natural convexification operator of the metric space concerned. This function is then used to construct a weighted mean-set that generalizes the notion of convex combination (CC) mean in the sense of Ter{\'a}n and Molchanov, the mean-set concept according to Mosina and the ordinary notion of $ k $-means ($k\geq 2$) of independent identically distributed (i.i.d.) random elements of the metric space. Two numerical examples are given for the cases when the metric space $ X= [0,1]$ and $ X=\mathbf{R}^{2} $. Moreover, an analogue of the Strong Law of Large Numbers (SLLN), the consistency problem and the Chebyshev's inequality for NCCCP spaces are established.