A uniform functional law of the logarithm for the local empirical process

We prove a uniform functional law of the logarithm for the local empirical process. To accomplish this we combine techniques from classical and abstract empirical process theory, Gaussian distributional approximation and probability on Banach spaces. The body of techniques we develop should prove useful to the study of the strong consistency of d-variate kernel-type nonparametric function estimators.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000024

Business angel investing

Business angels are conventionally defined as high net worth individuals who invest their own money, along with their time and expertise, directly in unquoted companies in which they have no family connection, in the hope of financial gain. The term angel was coined by Broadway insiders in the early 1900s to describe wealthy theatre-goers who made high risk investments in theatrical productions. Angels invested in these shows primarily for the privilege of rubbing shoulders with the theatre personalities that they admired. The term business angel was given to those individuals who perform essentially the same function in a business context (Benjamin and Margulis, 2000: 5). There is a long tradition of angel investing in businesses (Sohl, 2003). Moreover, angel investing is now an international phenomenon, found in all developed economies and now diffusing to emerging economies such as China (Lui Tingchi, and Chen Po Chang,, 2007). However, it has only attracted the attention of researchers since the 1980s

Using Technology as a Vehicle to Appropriately Integrate Mathematics and Science Instruction for the Middle School

At the College of William and Mary, pre-service middle school science and mathematics teachers enroll in their respective methods courses taught in the same time period. Both instructors emphasize the importance of the content pedagogy unique to their disciplines in their individual courses such as strategies for teaching problem solving, computation, proportional reasoning, algebraic and geometric thinking in mathematics, and strategies for teaching students how to investigate or design and conduct experiments in science. However, the two classes come together for sessions in which they examine the relationship of the two disciplines and the proper role of technology, both graphing calculator and computer, in their instruction Starting with resources such as Science in Seconds for Kids by Jean Potter [1], the science students collaborate with the math students to design and conduct brief experiments. The data generated is analyzed using spreadsheets and later graphing calculators. Various classes of mathematical curves are examined using data generated by sensors/probes and CBLs. Through this experience the pre-service teachers learn to work collaboratively with their colleagues on meaningful tasks, strengthening the effectiveness of all participants

CELSS scenario analysis: Breakeven calculations

A model of the relative mass requirements of food production components in a controlled ecological life support system (CELSS) based on regenerative concepts is described. Included are a discussion of model scope, structure, and example calculations. Computer programs for cultivar and breakeven calculations are also included

Experimental Design at the Intersection of Mathematics, Science, and Technology in Grades K-6

Interdisciplinary courses, highlighting as they do the area(s) the disciplines have in common, often give the misperception of a single body of knowledge and/or way of knowing. However, discipline based courses often leave the equally mistaken notion that the disciplines have nothing in common. The task of the methods courses described in this paper is to reach an appropriate balance so that our pre-service elementary (K-6) teachers have a realistic perception of the independence and interdependence of mathematics and science. At the College of William and Mary each cohort of pre-service elementary teachers enrolls in mathematics and science methods courses taught in consecutive hours. Both instructors emphasize the importance of the content pedagogy unique to their disciplines such as strategies for teaching problem solving, computation, algebraic thinking, and proportional reasoning in mathematics and strategies for teaching students how to investigate and understand the concepts of science. The instructors model interdisciplinary instruction by collaboratively teaching common content pedagogy such as the use of technology, data analysis, and interpretation. Students also identify real-life application of the mathematical principles they are learning that can be applied to science. The concept of simultaneously teaching appropriately selected math and science skills are stressed. Given this approach students are not left with the notion that mathematics is the handmaid of science nor the notion that it is the queen of the sciences. Rather, they view mathematics as a co-equal partner

On the Breiman conjecture

Let $Y_{1},Y_{2},\ldots$ be positive, nondegenerate, i.i.d. $G$ random variables, and independently let $X_{1},X_{2},\ldots$ be i.i.d. $F$ random variables. In this note we show that whenever $\sum X_{i}Y_{i}/\sum Y_{i}$ converges in distribution to nondegenerate limit for some $F\in \mathcal{F}$, in a specified class of distributions $\mathcal{F}$, then $G$ necessarily belongs to the domain of attraction of a stable law with index less than 1. The class $\mathcal{F}$ contains those nondegenerate $X$ with a finite second moment and those $X$ in the domain of attraction of a stable law with index $1<\alpha <2$

Couplings and Strong Approximations to Time Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions

We define a time dependent empirical process based on $n$ i.i.d.~fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for this process.Comment: To appear in the Journal of Theoretical Probability. 37 pages. Corrected version. The results on quantile processes are taken out and it will appear elsewher

A note on a maximal Bernstein inequality

We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ304 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The limit distribution of ratios of jumps and sums of jumps of subordinators

Let $V_{t}$ be a driftless subordinator, and let denote $m_{t}^{(1)} \geq m_{t}^{(2)} \geq\ldots$ its jump sequence on interval $[0,t]$. Put $V_{t}^{(k)} = V_{t} - m_{t}^{(1)} - \ldots- m_{t}^{(k)}$ for the $k$-trimmed subordinator. In this note we characterize under what conditions the limiting distribution of the ratios $V_{t}^{(k)} / m_{t}^{(k+1)}$ and $m_{t}^{(k+1)} / m_{t}^{(k)}$ exist, as $t \downarrow0$ or $t \to\infty$.Comment: 14 page