8 research outputs found
Recommended from our members
EDITORIAL: MEMORIAL ISSUE FOR CHARLES STEIN
The Institute of Mathematical Statistics (IMS) Council approved a proposal from its Committee on Memorials to dedicate this issue of the Annals of Statistics to Charles M. Stein, who died in 2016 aged 96. This memorialisation is a reflection of Stein’s distinction as a mathematical statistician, whose work continues to have a profound impact on the discipline
Exchangeable pairs, switchings, and random regular graphs
We consider the distribution of cycle counts in a random regular graph, which
is closely linked to the graph's spectral properties. We broaden the asymptotic
regime in which the cycle counts are known to be approximately Poisson, and we
give an explicit bound in total variation distance for the approximation. Using
this result, we calculate limiting distributions of linear eigenvalue
functionals for random regular graphs.
Previous results on the distribution of cycle counts by McKay, Wormald, and
Wysocka (2004) used the method of switchings, a combinatorial technique for
asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and
demonstrates an interesting connection between the two techniques.Comment: Very minor changes; 23 page
Exchangeable pairs and Poisson approximation
This is a survey paper on Poisson approximation using Stein's method of
exchangeable pairs. We illustrate using Poisson-binomial trials and many
variations on three classical problems of combinatorial probability: the
matching problem, the coupon collector's problem, and the birthday problem.
While many details are new, the results are closely related to a body of work
developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry
Goldstein, and their collaborators. Some comparison with these other approaches
is offered.Comment: Published at http://dx.doi.org/10.1214/154957805100000096 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Sequential importance sampling for estimating expectations over the space of perfect matchings
This paper makes three contributions to estimating the number of perfect
matching in bipartite graphs. First, we prove that the popular sequential
importance sampling algorithm works in polynomial time for dense bipartite
graphs. More carefully, our algorithm gives a -approximation for
the number of perfect matchings of a -dense bipartite graph, using
samples. With size on
each side and for , a -dense bipartite graph
has all degrees greater than .
Second, practical applications of the algorithm requires many calls to
matching algorithms. A novel preprocessing step is provided which makes
significant improvements.
Third, three applications are provided. The first is for counting Latin
squares, the second is a practical way of computing the greedy algorithm for a
card guessing game with feedback, and the third is for stochastic block models.
In all three examples, sequential importance sampling allows treating practical
problems of reasonably large sizes
Eigenvalue fluctuations for random regular graphs
One of the major themes of random matrix theory is that many asymptotic
properties of traditionally studied distributions of random matrices are
universal. We probe the edges of universality by studying the spectral
properties of random regular graphs. Specifically, we prove limit theorems for
the fluctuations of linear spectral statistics of random regular graphs. We
find both universal and non-universal behavior. Our most important tool is
Stein's method for Poisson approximation, which we develop for use on random
regular graphs.
This is my Ph.D. thesis, based on joint work with Ioana Dumitriu, Elliot
Paquette, and Soumik Pal. For the most part, it's a mashed up version of
arXiv:1109.4094, arXiv:1112.0704, and arXiv:1203.1113, but some things in here
are improved or new. In particular, Chapter 4 goes into more detail on some of
the proofs than arXiv:1203.1113 and includes a new section. See Section 1.3 for
more discussion on what's new and who contributed to what.Comment: 103 pages; Ph.D. thesis at the University of Washington, 201