45,319 research outputs found
Local limit theorems and mod-phi convergence
We prove local limit theorems for mod-{\phi} convergent sequences of random
variables, {\phi} being a stable distribution. In particular, we give two new
proofs of a local limit theorem in the framework of mod-phi convergence: one
proof based on the notion of zone of control, and one proof based on the notion
of mod-{\phi} convergence in L1(iR). These new approaches allow us to identify
the infinitesimal scales at which the stable approximation is valid. We
complete our analysis with a large variety of examples to which our results
apply, and which stem from random matrix theory, number theory, combinatorics
or statistical mechanics.Comment: 35 pages. Version 2: improved presentation, in particular for the
examples in Section
On spectral disjointness of powers for rank-one transformations and M\"obius orthogonality
We study the spectral disjointness of the powers of a rank-one
transformation. For a large class of rank-one constructions, including those
for which the cutting and stacking parameters are bounded, and other examples
such as rigid generalized Chacon's maps and Katok's map, we prove that
different positive powers of the transformation are pairwise spectrally
disjoint on the continuous part of the spectrum. Our proof involves the
existence, in the weak closure of {U_T^k: k in Z}, of "sufficiently many"
analytic functions of the operator U_T. Then we apply these disjointness
results to prove Sarnak's conjecture for the (possibly non-uniquely ergodic)
symbolic models associated to these rank-one constructions: All sequences
realized in these models are orthogonal to the M\"obius function
On simplicial toric varieties of codimension 2
We describe classes of toric varieties of codimension 2 which are either
minimally defined by 3 binomial equations over any algebraically closed field,
or are set-theoretic complete intersections in exactly one positive
characteristic.Comment: Revised version. To appear in: Rendiconti dell'Istituto di Matematica
dell'Universita' di Trieste. Dedicated to the memory of Fabio Ross
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