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On extensions of typical group actions
For every countable abelian group we find the set of all its subgroups
() such that a typical measure-preserving -action on a standard
atomless probability space can be extended to a free
measure-preserving -action on . The description of all
such pairs was made in purely group terms, in the language of the
dual , and -actions with discrete spectrum. As an application, we
answer a question when a typical -action can be extended to a -action
with some dynamic property, or to a -action at all. In particular, we offer
first examples of pairs satisfying both is countable abelian, and
a typical -action is not embeddable in a -action.Comment: 30 page
Multifractal properties of typical convex functions
We study the singularity (multifractal) spectrum of continuous convex
functions defined on . Let be the set of points at which
has a pointwise exponent equal to . We first obtain general upper bounds
for the Hausdorff dimension of these sets , for all convex functions
and all . We prove that for typical/generic (in the sense of
Baire) continuous convex functions , one has for all and in addition, we obtain that the set is empty if . Also, when is typical,
the boundary of belongs to
Universal Sequential Outlier Hypothesis Testing
Universal outlier hypothesis testing is studied in a sequential setting.
Multiple observation sequences are collected, a small subset of which are
outliers. A sequence is considered an outlier if the observations in that
sequence are generated by an "outlier" distribution, distinct from a common
"typical" distribution governing the majority of the sequences. Apart from
being distinct, the outlier and typical distributions can be arbitrarily close.
The goal is to design a universal test to best discern all the outlier
sequences. A universal test with the flavor of the repeated significance test
is proposed and its asymptotic performance is characterized under various
universal settings. The proposed test is shown to be universally consistent.
For the model with identical outliers, the test is shown to be asymptotically
optimal universally when the number of outliers is the largest possible and
with the typical distribution being known, and its asymptotic performance
otherwise is also characterized. An extension of the findings to the model with
multiple distinct outliers is also discussed. In all cases, it is shown that
the asymptotic performance guarantees for the proposed test when neither the
outlier nor typical distribution is known converge to those when the typical
distribution is known.Comment: Proc. of the Asilomar Conference on Signals, Systems, and Computers,
2014. To appea
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