48,523 research outputs found
Auto-tail dependence coefficients for stationary solutions of linear stochastic recurrence equations and for GARCH(1,1)
We examine the auto-dependence structure of strictly stationary solutions of linear stochastic recurrence equations and of strictly stationary GARCH(1, 1) processes from the point of view of ordinary and generalized tail dependence coefficients. Since such processes can easily be of infinite variance, a substitute for the usual auto-correlation function is needed
Precise large deviations for dependent regularly varying sequences
We study a precise large deviation principle for a stationary regularly
varying sequence of random variables. This principle extends the classical
results of A.V. Nagaev (1969) and S.V. Nagaev (1979) for iid regularly varying
sequences. The proof uses an idea of Jakubowski (1993,1997) in the context of
centra limit theorems with infinite variance stable limits. We illustrate the
principle for \sv\ models, functions of a Markov chain satisfying a polynomial
drift condition and solutions of linear and non-linear stochastic recurrence
equations
Nonlocal Reductions of a Generalized Heisenberg Ferromagnet Equation
We study nonlocal reductions of coupled equations in dimensions of the
Heisenberg ferromagnet type. The equations under consideration are completely
integrable and have a Lax pair related to a linear bundle in pole gauge. We
describe the integrable hierarchy of nonlinear equations related to our system
in terms of generating operators. We present some special solutions associated
with four distinct discrete eigenvalues of scattering operator. Using the Lax
pair diagonalization method, we derive recurrence formulas for the conserved
densities and find the first two simplest conserved densities.Comment: 14 pages. arXiv admin note: substantial text overlap with
arXiv:1711.0635
NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions
This article describes the implementation in the software package NumGfun of
classical algorithms that operate on solutions of linear differential equations
or recurrence relations with polynomial coefficients, including what seems to
be the first general implementation of the fast high-precision numerical
evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our
descriptions contain improvements over existing algorithms. We also provide
references to relevant ideas not currently used in NumGfun
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