Martingale-coboundary representation for stationary random fields

2017 ◽  
Vol 18 (02) ◽  
pp. 1850011 ◽  
Author(s):  
Dalibor Volný
Keyword(s):  
Large Deviations ◽  
Invariance Principle ◽  
Random Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
Dimension One ◽  
Necessary And Sufficient

We prove a martingale-coboundary representation for random fields with a completely commuting filtration. For random variables in [Formula: see text], we present a necessary and sufficient condition which is a generalization of Heyde’s condition for one-dimensional processes from 1975. For [Formula: see text] spaces with [Formula: see text] we give a necessary and sufficient condition which extends Volný’s result from 1993 to random fields and improves condition of El Machkouri and Giraudo from 2016. A new sufficient condition is presented which for dimension one improves Gordin’s condition from 1969. In application, new weak invariance principle and estimates of large deviations are found.

scholarly journals Invariance principle via orthomartingale approximation

2018 ◽  
Vol 18 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Davide Giraudo
Keyword(s):  
Invariance Principle ◽  
Random Fields ◽  
Partial Sums ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stationary Random Fields ◽  
Necessary And Sufficient

We obtain a necessary and sufficient condition for the orthomartingale-coboundary decomposition. We establish a sufficient condition for the approximation of the partial sums of a strictly stationary random fields by those of stationary orthomartingale differences. This condition can be checked under multidimensional analogues of the Hannan condition and the Maxwell–Woodroofe condition.

On the weak invariance principle for non-adapted stationary random fields under projective criteria

2021 ◽  
Author(s):  
Han-Mai Lin
Keyword(s):  
Invariance Principle ◽  
Random Fields ◽  
Long Memory ◽  
Sufficient Conditions ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
The Central Limit Theorem ◽  
Projective Criteria ◽  
Stationary Random Fields ◽  
Do So

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random fields non-necessarily adapted, under different normalizations. To do so, we first state sufficient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP hold. These projective criteria are in the spirit of Hannan’s condition and are well adapted to linear random fields with ortho-martingale innovations and which exhibit long memory.

scholarly journals Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sandro Zagatti
Keyword(s):  
Existence Result ◽  
Variational Problems ◽  
Convex Envelope ◽  
Minimum Problem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
First Order ◽  
Existence Of Minimizers ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>We study the minimum problem for functionals of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \mathcal{F}(u) = \int_{I} f(x, u(x), u^ \prime(x), u^ {\prime\prime}(x))\,dx, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where the integrand <inline-formula><tex-math id="M1">\begin{document}$ f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \mathbb{R} $\end{document}</tex-math></inline-formula> is not convex in the last variable. We provide an existence result assuming that the lower convex envelope <inline-formula><tex-math id="M2">\begin{document}$ \overline{f} = \overline{f}(x,p,q,\xi) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula> with respect to <inline-formula><tex-math id="M4">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is regular and enjoys a special dependence with respect to the i-th single components <inline-formula><tex-math id="M5">\begin{document}$ p_i, q_i, \xi_i $\end{document}</tex-math></inline-formula> of the vector variables <inline-formula><tex-math id="M6">\begin{document}$ p,q,\xi $\end{document}</tex-math></inline-formula>. More precisely, we assume that it is monotone in <inline-formula><tex-math id="M7">\begin{document}$ p_i $\end{document}</tex-math></inline-formula> and that it satisfies suitable affinity properties with respect to <inline-formula><tex-math id="M8">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> on the set <inline-formula><tex-math id="M9">\begin{document}$ \{f&gt; \overline{f}\} $\end{document}</tex-math></inline-formula> and with respect to <inline-formula><tex-math id="M10">\begin{document}$ q_i $\end{document}</tex-math></inline-formula> on the whole domain. We adopt refined versions of the integro-extremality method, extending analogous results already obtained for functionals with first order lagrangians. In addition we show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the solvability of this class of variational problems.</p>

On the stability of some solutions of the Stefan problem

1991 ◽  
Vol 2 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Riccardo Ricci ◽  
Xie Weiqing
Keyword(s):  
Stefan Problem ◽  
Travelling Wave ◽  
Sufficient Condition ◽  
Travelling Wave Solutions ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Necessary And Sufficient ◽  
Travelling Wave Front

We investigate the stability of travelling wave solutions of the one-dimensional under-cooled Stefan problem. We find a necessary and sufficient condition on the initial datum under which the free boundary is asymptotic to a travelling wave front. The method applies also to other types of solutions.

scholarly journals On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Rossitza Semerdjieva
Keyword(s):  
Free Boundary ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Sufficient Condition ◽  
Free Boundary Value Problem ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
Necessary And Sufficient ◽  
Infinite Differentiability ◽  
Differential Condition

AbstractWe consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on Cm-smoothness of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.

scholarly journals Existence of Cocompact Lattices in Lie Groups With a Bi-invariant Metric of Index 2

2020 ◽  
Author(s):  
Ines Kath
Keyword(s):  
Lie Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Invariant Metric ◽  
One Dimensional ◽  
Salem Numbers ◽  
Index 2 ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Cocompact Lattices

Abstract We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature $(2,n-2)$. We assume in addition that the Lie groups under consideration are simply-connected, indecomposable, and solvable. Then their centre is one- or two-dimensional. In both cases, a parametrisation of the set of such Lie groups is known. We give a necessary and sufficient condition for the existence of a lattice in terms of these parameters. For groups with one-dimensional centre this problem is related to Salem numbers.

On a theorem of van der Corput on uniform distribution

1957 ◽  
Vol 53 (4) ◽  
pp. 781-789 ◽  
Author(s):  
W. J. Coles
Keyword(s):  
Dimensional Space ◽  
Sufficient Condition ◽  
General Criterion ◽  
Two Dimensional ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
Quantitative Form ◽  
The One ◽  
Sufficiency Part ◽  
Necessary And Sufficient

Van der Corput has shown (2), using a general criterion of Weyl (1), that a necessary and sufficient condition, that a sequence of points Pn = (αn, βn) (n = 1, 2,…) in two-dimensional space be uniformly distributed modulo 1, is that for all pairs of integers (u, v) other than u = v = 0 the one-dimensional sequence (uαn + vβn) (n = 1, 2,…) is uniformly distributed modulo 1. The object of this paper is to give a quantitative form to the sufficiency part of this qualitative criterion.

Sign in / Sign up

Export Citation Format

Share Document