54 research outputs found
On the almost sure central limit theorem for ARX processes in adaptive tracking
The goal of this paper is to highlight the almost sure central limit theorem
for martingales to the control community and to show the usefulness of this
result for the system identification of controllable ARX(p,q) process in
adaptive tracking. We also provide strongly consistent estimators of the even
moments of the driven noise of a controllable ARX(p,q) process as well as
quadratic strong laws for the average costs and estimation errors sequences.
Our theoretical results are illustrated by numerical experiments
Small deviations of iterated processes in space of trajectories
We derive logarithmic asymptotics of probabilities of small deviations for
iterated processes in the space of trajectories. We find conditions under which
these asymptotics coincide with those of processes generating iterated
processes. When these conditions fail the asymptotics are quite different
Combinatorics of Feynman Diagrams for the Problems with Gaussian Random Field
The algorithm to calculate the generating function for the number of
``skeleton'' diagrams for the irreducible self-energy and vertex parts is
derived for the problems with Gaussian random fields. We find an exact
recurrence relation determining the number of diagrams for any given order of
perturbation theory, as well as its asymptotics for the large order limit.
These results are applied to the analysis of the problem of an electron in the
Gaussian random field with the ``white-noise'' correlation function. Assuming
the equality of all ``skeleton'' diagrams for the self-energy part in the given
order of perturbation theory, we construct the closed integral equation for the
one-particle Green's function, with its kernel defined by the previously
introduced generating function. Our analysis demonstrate that this
approximation gives the qualitatively correct form of the localized states
``tail'' in the density of states in the region of negative energies and is
apparently quite satisfactory in the most interesting region of strong
scattering close to the former band-edge, where we can derive the asymptotics
of the Green's function and density of states in the limit of very strong
scattering.Comment: 23 pages, 6 figures, RevTeX 3.0, Postscript figures attached,
Submitted to JET
Analytical realization of finite-size scaling for Anderson localization. Does the band of critical states exist for d>2?
An analytical realization is suggested for the finite-size scaling algorithm
based on the consideration of auxiliary quasi-1D systems. Comparison of the
obtained analytical results with the results of numerical calculations
indicates that the Anderson transition point is splitted into the band of
critical states. This conclusion is supported by direct numerical evidence
(Edwards and Thouless, 1972; Last and Thouless, 1974; Schreiber, 1985; 1990).
The possibility of restoring the conventional picture still exists but requires
a radical reinterpretetion of the raw numerical data.Comment: PDF, 11 page
Divergent Perturbation Series
Various perturbation series are factorially divergent. The behavior of their
high-order terms can be found by Lipatov's method, according to which they are
determined by the saddle-point configurations (instantons) of appropriate
functional integrals. When the Lipatov asymptotics is known and several lowest
order terms of the perturbation series are found by direct calculation of
diagrams, one can gain insight into the behavior of the remaining terms of the
series. Summing it, one can solve (in a certain approximation) various
strong-coupling problems. This approach is demonstrated by determining the
Gell-Mann - Low functions in \phi^4 theory, QED, and QCD for arbitrary coupling
constants. An overview of the mathematical theory of divergent series is
presented, and interpretation of perturbation series is discussed. Explicit
derivations of the Lipatov asymptotic forms are presented for some basic
problems in theoretical physics. A solution is proposed to the problem of
renormalon contributions, which hampered progress in this field in the late
1970s. Practical schemes for summation of perturbation series are described for
a coupling constant of order unity and in the strong-coupling limit. An
interpretation of the Borel integral is given for 'non-Borel-summable' series.
High-order corrections to the Lipatov asymptotics are discussed.Comment: Review article, 45 pages, PD
- …