4 research outputs found
Extreme value distributions and Renormalization Group
In the classical theorems of extreme value theory the limits of suitably
rescaled maxima of sequences of independent, identically distributed random
variables are studied. So far, only affine rescalings have been considered. We
show, however, that more general rescalings are natural and lead to new limit
distributions, apart from the Gumbel, Weibull, and Fr\'echet families. The
problem is approached using the language of Renormalization Group
transformations in the space of probability densities. The limit distributions
are fixed points of the transformation and the study of the differential around
them allows a local analysis of the domains of attraction and the computation
of finite-size corrections.Comment: 16 pages, 5 figures. Final versio
A Primer on Reproducing Kernel Hilbert Spaces
Reproducing kernel Hilbert spaces are elucidated without assuming prior
familiarity with Hilbert spaces. Compared with extant pedagogic material,
greater care is placed on motivating the definition of reproducing kernel
Hilbert spaces and explaining when and why these spaces are efficacious. The
novel viewpoint is that reproducing kernel Hilbert space theory studies
extrinsic geometry, associating with each geometric configuration a canonical
overdetermined coordinate system. This coordinate system varies continuously
with changing geometric configurations, making it well-suited for studying
problems whose solutions also vary continuously with changing geometry. This
primer can also serve as an introduction to infinite-dimensional linear algebra
because reproducing kernel Hilbert spaces have more properties in common with
Euclidean spaces than do more general Hilbert spaces.Comment: Revised version submitted to Foundations and Trends in Signal
Processin
Computing Limits of Sequences
So far, it seems that more emphasis was laid on the treatment of continuous functions, and although some mathematical fundaments are already available [1, 2], algorithmic approaches for the discrete case seem rare