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

Leave Vacant or Rent: The Optimal Decision for Absentee Home Sellers

Home-sellers may not occupy their property when listed for sale. While previous research has analyzed the effect of a vacancy on the sales price of residential properties, no research to date has quantified the economic benefits and costs of renting vis-à-vis leaving a property vacant while listed for sale. Renting a property will produce revenue but perhaps at the cost of a lower sale price and/or longer time on the market. This paper employs data on 55,202 homes sold in the Las Vegas, Nevada area to determine if renting a property increases or decreases the wealth position of home sellers. The results of an empirical test of the model are used to quantify the wealth effect. The findings indicate that renting the property produces a significant reduction in the home seller’s wealth position.

The Effect of Tax Laws and the Cost of Capital on the Size of Newly Constructed Strip Shopping Centers

While the impact of tax policy and other economic variables on the total amount of construction has been widely studied, this paper proposes that these variables also affect the size distribution of the properties constructed. The basic intuition is that there is a lower bound to the economically feasible size of a project due to economies of scale in construction. Events favorable to construction, such as lower interest rates and more favorable tax treatment, relax this lower bound permitting the construction of smaller properties. We test this proposition using data on newly constructed neighborhood shopping centers in Clark County, Nevada during the period from 1971 to 1999.