Uniform convergence of Vapnik--Chervonenkis classes under ergodic sampling

We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$, the relative frequencies of sets $C\in\mathcal{C}$ converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of $\mathcal{C}$. The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets.Comment: Published in at http://dx.doi.org/10.1214/09-AOP511 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Binomial-coefficient multiples of irrationals

Denote by $x$ a random infinite path in the graph of Pascal's triangle (left and right turns are selected independently with fixed probabilities) and by $d_n(x)$ the binomial coefficient at the $n$'th level along the path $x$. Then for a dense $G_{\delta}$ set of $\theta$ in the unit interval, $\{d_n(x)\theta \}$ is almost surely dense but not uniformly distributed modulo 1.Comment: 10 pages, to appear in Monatshefte f. Mat

Uniform approximation of Vapnik–Chervonenkis classes

For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik–Chervonenkis (VC) dimension or (ii) for every $\varepsilon >0$ there is a finite partition $\pi$ such the essential $\pi$-boundary of each set has measure at most $\varepsilon$. Immediate corollaries include the fact that a separable family with finite VC dimension has finite bracketing numbers, and satisfies uniform laws of large numbers for every ergodic process. From these corollaries, we derive analogous results for VC major and VC graph families of functions

BRAIN Initiative: Cutting-Edge Tools and Resources for the Community.

The overarching goal of the NIH BRAIN (Brain Research through Advancing Innovative Neurotechnologies) Initiative is to advance the understanding of healthy and diseased brain circuit function through technological innovation. Core principles for this goal include the validation and dissemination of the myriad innovative technologies, tools, methods, and resources emerging from BRAIN-funded research. Innovators, BRAIN funding agencies, and non-Federal partners are working together to develop strategies for making these products usable, available, and accessible to the scientific community. Here, we describe several early strategies for supporting the dissemination of BRAIN technologies. We aim to invigorate a dialogue with the neuroscience research and funding community, interdisciplinary collaborators, and trainees about the existing and future opportunities for cultivating groundbreaking research products into mature, integrated, and adaptable research systems. Along with the accompanying Society for Neuroscience 2019 Mini-Symposium, "BRAIN Initiative: Cutting-Edge Tools and Resources for the Community," we spotlight the work of several BRAIN investigator teams who are making progress toward providing tools, technologies, and services for the neuroscience community. These tools access neural circuits at multiple levels of analysis, from subcellular composition to brain-wide network connectivity, including the following: integrated systems for EM- and florescence-based connectomics, advances in immunolabeling capabilities, and resources for recording and analyzing functional connectivity. Investigators describe how the resources they provide to the community will contribute to achieving the goals of the NIH BRAIN Initiative. Finally, in addition to celebrating the contributions of these BRAIN-funded investigators, the Mini-Symposium will illustrate the broader diversity of BRAIN Initiative investments in cutting-edge technologies and resources