1,306 research outputs found
Low-degree tests at large distances
We define tests of boolean functions which distinguish between linear (or
quadratic) polynomials, and functions which are very far, in an appropriate
sense, from these polynomials. The tests have optimal or nearly optimal
trade-offs between soundness and the number of queries.
In particular, we show that functions with small Gowers uniformity norms
behave ``randomly'' with respect to hypergraph linearity tests.
A central step in our analysis of quadraticity tests is the proof of an
inverse theorem for the third Gowers uniformity norm of boolean functions.
The last result has also a coding theory application. It is possible to
estimate efficiently the distance from the second-order Reed-Muller code on
inputs lying far beyond its list-decoding radius
Null flows, positive flows and the structure of stationary symmetric stable processes
This paper elucidates the connection between stationary symmetric
alpha-stable processes with 0<alpha<2 and nonsingular flows on measure spaces
by describing a new and unique decomposition of stationary stable processes
into those corresponding to positive flows and those corresponding to null
flows. We show that a necessary and sufficient for a stationary stable process
to be ergodic is that its positive component vanishes
Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes
We study the partial maxima of stationary \alpha-stable processes. We relate
their asymptotic behavior to the ergodic theoretical properties of the flow. We
observe a sharp change in the asymptotic behavior of the sequence of partial
maxima as flow changes from being dissipative to being conservative, and argue
that this may indicate a change from a short memory process to a long memory
process.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000026
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