Mixing in Non-Quasirandom Groups

We initiate a systematic study of mixing in non-quasirandom groups. Let A and B be two independent, high-entropy distributions over a group G. We show that the product distribution AB is statistically close to the distribution F(AB) for several choices of G and F, including: 1) G is the affine group of 2x2 matrices, and F sets the top-right matrix entry to a uniform value, 2) G is the lamplighter group, that is the wreath product of ?? and ?_{n}, and F is multiplication by a certain subgroup, 3) G is H? where H is non-abelian, and F selects a uniform coordinate and takes a uniform conjugate of it. The obtained bounds for (1) and (2) are tight. This work is motivated by and applied to problems in communication complexity. We consider the 3-party communication problem of deciding if the product of three group elements multiplies to the identity. We prove lower bounds for the groups above, which are tight for the affine and the lamplighter groups

The Poset of Hypergraph Quasirandomness

Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's.Comment: 43 pages, 1 figur

Constructing Sobol' sequences with better two-dimensional projections

Direction numbers for generating Sobol' sequences that satisfy the so-called Property A in up to 1111 dimensions have previously been given in Joe and Kuo [ACM Trans. Math. Software, 29 (2003), pp. 49–57]. However, these Sobol' sequences may have poor two-dimensional projections. Here we provide a new set of direction numbers alleviating this problem. These are obtained by treating Sobol' sequences in d dimensions as (t, d)-sequences and then optimizing the t-values of the two-dimensional projections. Our target dimension is 21201

A Censored Random Coefficients Model for Pooled Survey Data with Application to the Estimation of Power Outage Costs

In many surveys multiple observations on the dependent variable are collected from a given respondent. The resulting pooled data set is likely to be censored and to exhibit cross-sectional heterogeneity. We propose a model that addresses both issues by allowing regression coefficients to vary randomly across respondents and by using the Geweke-Hajivassiliou-Keane simulator and Halton sequences to estimate high-order probabilities. We show how this framework can be usefully applied to the estimation of power outage costs to firms using data from a recent survey conducted by a U.S. utility. Our results strongly reject the hypotheses of parameter constancy and cross-sectional homogeneity.