Statistics Seminars: Realizability problem for point random fields
20 February 2012 14:00 in CM221
To reconstruct in a systematical way from observable quantities, the underlying effective description of a complex system on relevant scales is a task of enormous practical relevance. Realizability considers the partial question if the system can be described by point-like objects on the relevant scale, cf. Percus(1964) and Crawford et al. (2003).
In this talk, the realizability problem is introduced and identified as an infinite dimensional version of the classical truncated power moment problem. One can associate a linear functional on the space of polynomials to any kind of moment problem. A classical theorem for complete moment sequences, see e.g. Haviland(1935/6) states that solvability of the moment problem is equivalent to positivity of this functional. However, this is wrong in general for truncated moment problems. A new general approach for truncated moment problems will be presented which overcomes this difficulty. To our knowledge this approach is also new for finite dimensional problem, however it may be more adapted for infinite dimensional problems.
Moment problems are very difficult in more than one dimension and little is known in general. We will give explicit solutions for the moment problem in particular regimes and study further properties of the solutions.
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