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 angels

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

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

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

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

Uniform in bandwidth consistency of kernel-type function estimators

We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya-Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.Comment: Published at http://dx.doi.org/10.1214/009053605000000129 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Randomly Weighted Self-normalized L\'evy Processes

Let $(U_t,V_t)$ be a bivariate L\'evy process, where $V_t$ is a subordinator and $U_t$ is a L\'evy process formed by randomly weighting each jump of $V_t$ by an independent random variable $X_t$ having cdf $F$. We investigate the asymptotic distribution of the self-normalized L\'evy process $U_t/V_t$ at 0 and at $\infty$. We show that all subsequential limits of this ratio at 0 ($\infty$) are continuous for any nondegenerate $F$ with finite expectation if and only if $V_t$ belongs to the centered Feller class at 0 ($\infty$). We also characterize when $U_t/V_t$ has a non-degenerate limit distribution at 0 and $\infty$.Comment: 32 page