scholarly journals A new proof of the dimension gap for the Gauss map

2021 ◽  
pp. 1-29
Author(s):  
NATALIA JURGA
Keyword(s):  
Lower Bounds ◽  
Asymptotic Variance ◽  
Thermodynamic Formalism ◽  
Gauss Map ◽  
Probability Vector ◽  
Bernoulli Measure ◽  
Bernoulli Measures

Abstract In [4], Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map T(x)=1/x mod 1 satisfy a ‘dimension gap’ meaning that for some c > 0, sup p dim μ p < 1– c, where μp denotes the (pushforward) Bernoulli measure for the countable probability vector p. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.

Nuisance parameters, mixture models, and the efficiency of partial likelihood estimators

1980 ◽  
Vol 296 (1427) ◽  
pp. 639-662 ◽  
Keyword(s):  
Lower Bounds ◽  
Statistical Models ◽  
Asymptotic Variance ◽  
Partial Likelihood ◽  
Nuisance Parameters ◽  
Direct Proportion ◽  
Likelihood Estimator ◽  
Unbiased Estimators ◽  
Asymptotically Normal ◽  
Independent Observations

This paper establishes lower bounds for estimation in parametric statistical models in which one wishes to estimate a real-valued parameter of interest in the presence of nuisance parameters which are accruing in number in direct proportion to the number of independent observations. The formal setting requires that the nuisance parameters be independent observations from an unknown distribution. In this setting an information measure analogous to the Fisher information is derived. It is then used to generate lower bounds for the variance of unbiased estimators and also for the asymptotic variance of consistent asymptotically normal estimators. Under certain conditions, consistent asymptotically normal estimators can be generated by maximizing factors of the complete likelihood, even though the maximum likelihood estimator is inconsistent. These estimators can be fully efficient in the sense of meeting the lower bounds despite their apparent wasteful use of the likelihood, as is demonstrated, in several important examples, by the use of a natural sufficient condition.

Approximation of Bernoulli measures for non-uniformly hyperbolic systems

2018 ◽  
Vol 40 (1) ◽  
pp. 233-247
Author(s):  
GANG LIAO ◽  
WENXIANG SUN ◽  
EDSON VARGAS ◽  
SHIROU WANG
Keyword(s):  
Invariant Measure ◽  
Bernoulli Shift ◽  
Hyperbolic Systems ◽  
Weak Mixing ◽  
Bernoulli Measure ◽  
Uniformly Hyperbolic Systems ◽  
Bernoulli Measures

An invariant measure is called a Bernoulli measure if the corresponding dynamics is isomorphic to a Bernoulli shift. We prove that for$C^{1+\unicode[STIX]{x1D6FC}}$diffeomorphisms any weak mixing hyperbolic measure could be approximated by Bernoulli measures. This also holds true for$C^{1}$diffeomorphisms preserving a weak mixing hyperbolic measure with respect to which the Oseledets decomposition is dominated.

scholarly journals A Note on the Generalized -Bernoulli Measures with Weight

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
T. Kim ◽  
S. H. Lee ◽  
D. V. Dolgy ◽  
C. S. Ryoo
Keyword(s):  
Bernoulli Numbers ◽  
Bernoulli Measure ◽  
Bernoulli Measures

We discuss a new concept of the -extension of Bernoulli measure. From those measures, we derive some interesting properties on the generalized -Bernoulli numbers with weight attached to .

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