Laws of large numbers in stochastic geometry with statistical applications

Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $\nu_n=\sum_i\xi_i$, where $\xi_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $\xi_i$ is determined by the (suitably rescaled) set of points near $X_i$. For bounded test functions $f$ on $R^d$, we give weak and strong laws of large numbers for $\nu_n(f)$. The general results are applied to demonstrate that an unknown set $A$ in $d$-space can be consistently estimated, given data on which of the points $X_i$ lie in $A$, by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ5167 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Liberty University\u27s Lawyering Skills Program: Integrating Legal Theory in a Practice-Oriented Curriculum

Law schools are not preparing lawyers for the practice of law. While modern legal education may teach analytical reasoning, skills training continues to suffer. The lawyering skills program developed by the Liberty University School of Law addresses the need to train lawyers in the practice of law. When seeking to build a top quality law school, Liberty University took seriously the challenge to address the void in legal education, particularly with respect to lawyering skills. The foundational principles of law are infused into the legal curriculum and lawyering skills program. The required substantive law courses are structured to have a relationship with lawyering skills so that they mutually reinforce each other

Apportionment and Contribution of Workers\u27 Compensation Benefits

The apportionment of benefits between a claimant and a carrier and contribution of benefits between multiple carriers has been a confusing area of law which has generated conflicting appellate court opinions. This article will explore the differences between Florida Statutes sec. 44012(5)(a) in sec. 440.42(3). After discussing the differences, this article will then focus on the multiple applications of sec. 440.42(3), the section dealing with the contribution of responsibility between carriers

Poisson process Fock space representation, chaos expansion and covariance inequalities

We consider a Poisson process $\eta$ on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of $\eta$. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener-Ito chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincare inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris-FKG-inequalities for monotone functions of $\eta$.Comment: 25 page