10,230 research outputs found
Arithmetic structures in random sets
We extend two well-known results in additive number theory, S\'ark\"ozy's
theorem on square differences in dense sets and a theorem of Green on long
arithmetic progressions in sumsets, to subsets of random sets of asymptotic
density 0. Our proofs rely on a restriction-type Fourier analytic argument of
Green and Green-Tao.Comment: 22 page
Random sets and exact confidence regions
An important problem in statistics is the construction of confidence regions
for unknown parameters. In most cases, asymptotic distribution theory is used
to construct confidence regions, so any coverage probability claims only hold
approximately, for large samples. This paper describes a new approach, using
random sets, which allows users to construct exact confidence regions without
appeal to asymptotic theory. In particular, if the user-specified random set
satisfies a certain validity property, confidence regions obtained by
thresholding the induced data-dependent plausibility function are shown to have
the desired coverage probability.Comment: 14 pages, 2 figure
Sums and differences of correlated random sets
Many fundamental questions in additive number theory (such as Goldbach's
conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be
expressed in the language of sum and difference sets. As a typical pair of
elements contributes one sum and two differences, we expect that for a finite set . However, in 2006 Martin and O'Bryant showed that a
positive proportion of subsets of are sum-dominant, and Zhao
later showed that this proportion converges to a positive limit as . Related problems, such as constructing explicit families of
sum-dominant sets, computing the value of the limiting proportion, and
investigating the behavior as the probability of including a given element in
to go to zero, have been analyzed extensively.
We consider many of these problems in a more general setting. Instead of just
one set , we study sums and differences of pairs of \emph{correlated} sets
. Specifically, we place each element in with
probability , while goes in with probability if
and probability if . If , we
call the pair a \emph{sum-dominant -pair}. We prove
that for any fixed in , is a
sum-dominant -pair with positive probability, and show that
this probability approaches a limit . Furthermore, we show that
the limit function is continuous. We also investigate what
happens as decays with , generalizing results of Hegarty-Miller on phase
transitions. Finally, we find the smallest sizes of MSTD pairs.Comment: Version 1.0, 19 pages. Keywords: More Sum Than Difference sets,
correlated random variables, phase transitio
Countable Random Sets: Uniqueness in Law and Constructiveness
The first part of this article deals with theorems on uniqueness in law for
\sigma-finite and constructive countable random sets, which in contrast to the
usual assumptions may have points of accumulation. We discuss and compare two
approaches on uniqueness theorems: First, the study of generators for
\sigma-fields used in this context and, secondly, the analysis of hitting
functions. The last section of this paper deals with the notion of
constructiveness. We will prove a measurable selection theorem and a
decomposition theorem for constructive countable random sets, and study
constructive countable random sets with independent increments.Comment: Published in Journal of Theoretical Probability
(http://www.springerlink.com/content/0894-9840/). The final publication is
available at http://www.springerlink.co
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