Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in $${\mathbb {Z}}^d$$

For an arbitrary transient random walk $$(S_n)_{n\ge 0}$$ ( S n ) n ≥ 0 in $${\mathbb {Z}}^d$$ Z d , $$d\ge 1$$ d ≥ 1 , we prove a strong law of large numbers for the spatial sum $$\sum _{x\in {\mathbb {Z}}^d}f(l(n,x))$$ ∑ x ∈ Z d f ( l ( n , x ) ) of a function f of the local times $$l(n,x)=\sum _{i=0}^n{\mathbb {I}}\{S_i=x\}$$ l ( n , x ) = ∑ i = 0 n I { S i = x } . Particular cases are the number of visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function $$f(i)={\mathbb {I}}\{i\ge 1\}$$ f ( i ) = I { i ≥ 1 } ; $$\alpha $$ α -fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to $$f(i)=i^\alpha $$ f ( i ) = i α ; sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where $$f(i)={\mathbb {I}}\{i=j\}$$ f ( i ) = I { i = j } .

UNIFORM BAHADUR REPRESENTATION FOR NONPARAMETRIC CENSORED QUANTILE REGRESSION: A REDISTRIBUTION-OF-MASS APPROACH

Censored quantile regressions have received a great deal of attention in the literature. In a linear setup, recent research has found that an estimator based on the idea of “redistribution-of-mass” in Efron (1967, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 831–853, University of California Press) has better numerical performance than other available methods. In this paper, this idea is combined with the local polynomial kernel smoothing for nonparametric quantile regression of censored data. We derive the uniform Bahadur representation for the estimator and, more importantly, give theoretical justification for its improved efficiency over existing estimation methods. We include an example to illustrate the usefulness of such a uniform representation in the context of sufficient dimension reduction in regression analysis. Finally, simulations are used to investigate the finite sample performance of the new estimator.

AN INTRODUCTION TO EXPLORING LAW, DISABILITY, AND THE CHALLENGE OF EQUALITY IN CANADA AND THE UNITED STATES: PAPERS FROM THE BERKELEY SYMPOSIUM

It brings me great pleasure to write this Introduction to Exploring Law, Disability, and the Challenge of Equality in Canada and the United States. This special collection of articles in the Windsor Yearbook of Access to Justice [WYAJ] stems from a symposium of the same name held at the Berkeley Law School at the University of California on 5 December 2014. Writing this introduction allows me to bring together my identities as a law and disability scholar, the principal organizer and convener of the Berkeley Symposium, and editor-in-chief of the WYAJ.In these roles, I have had the opportunity to engage with this set of articles and their authors in a distinct way – from the early versions of these articles through to the final peer-reviewed publications. The Berkeley Symposium is the first conference, of which we are aware, to bring together scholars and experts from both Canada and the United States to present research and exchange ideas on equality issues affecting persons with disabilities in both countries.1 Each academic was invited to write about an equality issue of their choice that is of contemporary concern to persons with disabilities, and to focus on Canada, the United States,or both, at their option. The result is a set of articles that is simultaneously introspective and comparative.

THE INTEGRATED MEAN SQUARED ERROR OF SERIES REGRESSION AND A ROSENTHAL HILBERT-SPACE INEQUALITY

This paper develops uniform approximations for the integrated mean squared error (IMSE) of nonparametric series regression estimators, including both least-squares and averaging least-squares estimators. To develop these approximations, we also generalize an important probability inequality of Rosenthal (1970, Israel Journal of Mathematics 8, 273–303; 1972, Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 149–167. University of California Press) to the case of Hilbert-space valued random variables.