Regularity Conditions for Stationary Processes

2003 ◽  
pp. 415-429
Author(s):  
Vladimir Peller
Keyword(s):  
Regularity Conditions ◽  
Stationary Processes

CAPABILITY MEASURES FOR m -DEPENDENT STATIONARY PROCESSES

Statistics ◽  
2003 ◽  
Vol 37 (1) ◽  
pp. 1-24 ◽  
Author(s):  
SY-MIEN CHEN ◽  
YU-SHENG HSU ◽  
W. L. PEARN
Keyword(s):  
Stationary Processes

Standard Asymptotic Theory

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Asymptotic Theory ◽  
Asymptotic Variance ◽  
Statistical Significance ◽  
Regularity Conditions ◽  
Parameter Estimates ◽  
Asymptotic Results ◽  
Ml Estimation ◽  
True Parameter ◽  
Log Likelihood ◽  
Ml Estimator

This book relies on maximum likelihood (ML) estimation of parameters. Asymptotic theory assumes regularity conditions hold when the ML estimator is consistent. Typically an additional third derivative condition is assumed to ensure that the ML estimator is also asymptotically normally distributed. Standard asymptotic results that then hold are summarized in this chapter; for example, the asymptotic variance of the ML estimator is then given by the Fisher information formula, and the log-likelihood ratio, the Wald and the score statistics for testing the statistical significance of parameter estimates are all asymptotically equivalent. Also, the useful profile log-likelihood then behaves exactly as a standard log-likelihood only in a parameter space of just one dimension. Further, the model can be reparametrized to make it locally orthogonal in the neighbourhood of the true parameter value. The large exponential family of models is briefly reviewed where a unified set of regular conditions can be obtained.

scholarly journals Kinetics of rare events for non-Markovian stationary processes and application to polymer dynamics

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
N. Levernier ◽  
O. Bénichou ◽  
R. Voituriez ◽  
T. Guérin
Keyword(s):  
Rare Events ◽  
Polymer Dynamics ◽  
Stationary Processes ◽  
Kinetics Of

scholarly journals Evidence for a 5d F-theorem

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Martin Fluder ◽  
Christoph F. Uhlemann
Keyword(s):  
Free Energy ◽  
Renormalization Group ◽  
General Expression ◽  
Geometric Parameters ◽  
Regularity Conditions ◽  
Large Sample ◽  
Large Numbers ◽  
Renormalization Group Flows

Abstract Renormalization group flows are studied between 5d SCFTs engineered by (p, q) 5-brane webs with large numbers of external 5-branes. A general expression for the free energy on S5 in terms of single-valued trilogarithm functions is derived from their supergravity duals, which are characterized by the 5-brane charges and additional geometric parameters. The additional geometric parameters are fixed by regularity conditions, and we show that the solutions to the regularity conditions extremize a trial free energy. These results are used to survey a large sample of $$ \mathcal{O} $$ O (105) renormalization group flows between different 5d SCFTs, including Higgs branch flows and flows that preserve the SU(2) R- symmetry. In all cases the free energy changes monotonically towards the infrared, in line with a 5d F -theorem.

scholarly journals Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

2021 ◽  
Author(s):  
Jan Bohr
Keyword(s):  
Stability Estimate ◽  
Scattering Data ◽  
Regularity Conditions ◽  
Uniform Bounds ◽  
Ray Transform ◽  
Shallow Layer ◽  
Matrix Potential ◽  
Novel Method ◽  
Statistical Consistency ◽  
Appl Math

AbstractNon-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.

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