1,737 research outputs found
The Monotone Cumulants
In the present paper we define the notion of generalized cumulants which
gives a universal framework for commutative, free, Boolean, and especially,
monotone probability theories. The uniqueness of generalized cumulants holds
for each independence, and hence, generalized cumulants are equal to the usual
cumulants in the commutative, free and Boolean cases. The way we define
(generalized) cumulants needs neither partition lattices nor generating
functions and then will give a new viewpoint to cumulants. We define ``monotone
cumulants'' in the sense of generalized cumulants and we obtain quite simple
proofs of central limit theorem and Poisson's law of small numbers in monotone
probability theory. Moreover, we clarify a combinatorial structure of
moment-cumulant formula with the use of ``monotone partitions''.Comment: 13 pages; minor changes and correction
Photon localization revisited
In the light of Newton-Wigner-Wightman theorem of localizability question, we
have proposed before a typical generation mechanism of effective mass for
photons to be localized in the form of polaritons owing to photon-media
interactions. In this paper, the general essence of this example model is
extracted in such a form as Quantum Field Ontology associated with
Eventualization Principle, which enables us to explain the mutual relations
back and forth, between quantum fields and various forms of particles in the
localized form of the former.Comment: arXiv admin note: substantial text overlap with arXiv:1101.578
A self-similar process arising from a random walk with random environment in random scenery
In this article, we merge celebrated results of Kesten and Spitzer [Z.
Wahrsch. Verw. Gebiete 50 (1979) 5-25] and Kawazu and Kesten [J. Stat. Phys. 37
(1984) 561-575]. A random walk performs a motion in an i.i.d. environment and
observes an i.i.d. scenery along its path. We assume that the scenery is in the
domain of attraction of a stable distribution and prove that the resulting
observations satisfy a limit theorem. The resulting limit process is a
self-similar stochastic process with non-trivial dependencies.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ234 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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