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 A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion

2016 ◽  
Vol 16 (03) ◽  
pp. 1660012 ◽  
Author(s):  
Ian Melbourne ◽  
Paulo Varandas
Keyword(s):  
Invariance Principle ◽  
Hyperbolic Systems ◽  
Return Time ◽  
Limit Laws ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
The Central Limit Theorem ◽  
Statistical Limit ◽  
Contraction And Expansion ◽  
Nonuniformly Hyperbolic

We provide a systematic approach for deducing statistical limit laws via martingale-coboundary decomposition, for nonuniformly hyperbolic systems with slowly contracting and expanding directions. In particular, if the associated return time function is square-integrable, then we obtain the central limit theorem, the weak invariance principle, and an iterated version of the weak invariance principle.

scholarly journals A weak invariance principle and asymptotic stability for evolution equations with bounded generators

1995 ◽  
Vol 18 (2) ◽  
pp. 255-264
Author(s):  
E. N. Chukwu ◽  
P. Smoczynski
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Equilibrium Point ◽  
Invariance Principle ◽  
Infinitesimal Generator ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Chemical Processes ◽  
Weak Invariance Principle ◽  
Weak Invariance

IfVis a Lyapunov function of an equationdu/dt=u′=Zuin a Banach space then asymptotic stability of an equilibrium point may be easily proved if it is known thatsup(V′)<0on sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is proved for a bounded infinitesimal generatorZunder a weaker assumptionV′≤0(which alone implies ordinary stability only) if some observability condition, involvingZand the Frechet derivative ofV′, is satisfied. The proof is based on an extension of LaSalle's invariance principle, which yields convergence in a weak topology and uses a strongly continuous Lyapunov function. The theory is illustrated with an example of an integro-differential equation of interest in the theory of chemical processes. In this case strong asymptotic stability is deduced from the weak one and explicit sufficient conditions for stability are given. In the case of a normal infinitesimal generatorZin a Hilbert space, strong asymptotic stability is proved under the following assumptionsZ*+Zis weakly negative definite andKer Z={0}. The proof is based on spectral theory.

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 Mixing rates for potentials of non-summable variations

2021 ◽  
pp. 1-18
Author(s):  
CHRISTOPHE GALLESCO ◽  
DANIEL Y. TAKAHASHI
Keyword(s):  
Invariance Principle ◽  
Type Inequality ◽  
Upper Bounds ◽  
Decay Of Correlations ◽  
Relaxation Rates ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
Coupling Inequality ◽  
Mixing Rates

Abstract Mixing rates, relaxation rates, and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations. This paper exhibits upper bounds for these quantities for dynamics defined by potentials with square-summable variations. We obtain these bounds as corollaries of a new block coupling inequality between pairs of dynamics starting with different histories. As applications of our results, we prove a new weak invariance principle and a Hoeffding-type inequality.

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