Related papers: Concentration of product spaces
In this book, we study Gromov's metric geometric theory on the space of metric measure spaces, based on the idea of concentration of measure phenomenon due to L\'evy and Milman. Although most of the details are omitted in the original…
We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with…
We explore the concentration properties of the ratio between the geometric mean and the arithmetic mean, showing that for certain sequences of weights one does obtain concentration, around a value that depends on the sequence.
Concentration of measure is studied, and obtained, for stable and related random vectors.
The concentration of measure prenomenon roughly states that, if a set $A$ in a product $\Omega^N$ of probability spaces has measure at least one half, ``most'' of the points of $\Omega^N$ are ``close'' to $A$. We proceed to a systematic…
We study the statistical convergence of metric valued sequences and of their subsequences. The interplay between the statistical and usual convergences in metric spaces is also studied.
In this paper some links between the density of a set of integers and the density of its sumset, product set and set of subset sums are presented.
A labeled metric space is intuitively speaking a metric space together with a special set of points to be understood as the geometric boundary of the space. We study basic properties of a recently introduced labeled Gromov-Hausdorff…
In this article we studied the relationship between metric spaces and multiplicative metric spaces. Also, we pointed out some fixed and common fixed point results under some contractive conditions in multiplicative metric spaces can be…
We describe some Cartesian products of metric spaces and find conditions under which products of ultrametric spaces are ultrametric.
We prove an apparently novel concentration of measure result for Markov tree processes. The bound we derive reduces to the known bounds for Markov processes when the tree is a chain, thus strictly generalizing the known Markov process…
The measure concentration property of an mm-space $X$ is roughly described as that any 1-Lipschitz map on $X$ to a metric space $Y$ is almost close to a constant map. The target space $Y$ is called the screen. The case of $Y=\mathbb{R}$ is…
We prove that the sequence of cones of metric measure spaces converges if the sequence of base spaces converges in Gromov's box, concentration, and weak topologies. As an application, we show that the generalized Cauchy distribution with…
We survey some basic results on the Gromov-Prohorov distance between metric measure spaces. (We do not claim any new results.) We give several different definitions and show the equivalence of them. We also show that convergence in the…
The phenomenon of concentration of measure on high dimensional structures is usually stated in terms of a metric space with a Borel measure, also called an mm-space. We extend some of the mm-space concepts to the setting of a quasi-metric…
Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several…
Concentration of measure is a phenomenon in which a random variable that depends in a smooth way on a large number of independent random variables is essentially constant. The random variable will "concentrate" around its median or…
We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all…
The purpose of this paper is to define statistically convergent sequences with respect to the metrics on generalized metric spaces (g-metric spaces) and investigate basic properties of this statistical form of convergence.
We study the limits of sequences of spheres and complex projective spaces with unbounded dimensions. A sequence of spheres (resp. complex projective spaces) either is a Levy family, infinitely dissipates, or converges to (resp. the Hopf…