Functional Chung laws for small increments of the empirical process and a lower bound in the strong invariance principle

2010 ◽  
Vol 61 (1-2) ◽  
pp. 67-102
Author(s):  
Philippe Berthet
Keyword(s):  
Lower Bound ◽  
Invariance Principle ◽  
Empirical Process ◽  
Strong Invariance ◽  
Strong Invariance Principle ◽  
Small Increments

scholarly journals Rates in the strong invariance principle for ergodic automorphisms of the torus

2014 ◽  
Vol 14 (02) ◽  
pp. 1350021
Author(s):  
Jérôme Dedecker ◽  
Florence Merlevède ◽  
Françoise Pène
Keyword(s):  
Invariance Principle ◽  
Fourier Coefficients ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Dimensional Torus ◽  
Strong Invariance ◽  
Ergodic Automorphism ◽  
Strong Invariance Principle ◽  
Natural Way

Let T be an ergodic automorphism of the d-dimensional torus 𝕋d. In the spirit of Le Borgne, we give conditions on the Fourier coefficients of a function f from 𝕋d to ℝ under which the partial sums f ◦ T + f ◦ T2 + ⋯ + f ◦ Tn satisfy a strong invariance principle. Next, reinforcing the condition on the Fourier coefficients in a natural way, we obtain explicit rates of convergence in the strong invariance principle, up to n1/4 log n.

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