Long memory is an important phenomenon that arises sometimes in the analysis of time series or spatial data.Most of the definitions concerning the long memory of a stationary process are based on the second-order prop...Long memory is an important phenomenon that arises sometimes in the analysis of time series or spatial data.Most of the definitions concerning the long memory of a stationary process are based on the second-order properties of the process.The mutual information between the past and future I_(p−f) of a stationary process represents the information stored in the history of the process which can be used to predict the future.We suggest that a stationary process can be referred to as long memory if its I_(p−f) is infinite.For a stationary process with finite block entropy,I_(p−f) is equal to the excess entropy,which is the summation of redundancies that relate the convergence rate of the conditional(differential)entropy to the entropy rate.Since the definitions of the I_(p−f) and the excess entropy of a stationary process require a very weak moment condition on the distribution of the process,it can be applied to processes whose distributions are without a bounded second moment.A significant property of I_(p−f) is that it is invariant under one-to-one transformation;this enables us to know the I_(p−f) of a stationary process from other processes.For a stationary Gaussian process,the long memory in the sense of mutual information is more strict than that in the sense of covariance.We demonstrate that the I_(p−f) of fractional Gaussian noise is infinite if and only if the Hurst parameter is H∈(1/2,1).展开更多
A Lorenz map f : I --> I is a one dimensional piecewise monotone map with a single discontinuity c. Let [GRAPHICS] be the collection of all the preimsges of c. Authors prove that if C'(f) is countable then ther...A Lorenz map f : I --> I is a one dimensional piecewise monotone map with a single discontinuity c. Let [GRAPHICS] be the collection of all the preimsges of c. Authors prove that if C'(f) is countable then there exists M such that Card(omega(x)) less than or equal to M for all x is an element of I. If C'(f) is uncountable then omega(x) is uncountable for some x is an element of I. So f is asymptotically periodic if and only if C'(f) is countable.展开更多
We consider the topological behaviors of continuous maps with one topological attractor on compact metric space X.This kind of map is a generalization of maps such as topologically expansive Lorenz map,unimodal map wi...We consider the topological behaviors of continuous maps with one topological attractor on compact metric space X.This kind of map is a generalization of maps such as topologically expansive Lorenz map,unimodal map without homtervals and so on.Under the finiteness and basin conditions,we provide a leveled A-R pair decomposition for such maps,and characterize α-limit set of each point.Based on weak Morse decomposition of X,we construct a bounded Lyapunov function V(x),which gives a clear description of orbit behavior of each point in X except a meager set.展开更多
基金supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry,the Key Scientific Research Project of Hunan Provincial Education Department (19A342)the National Natural Science Foundation of China (11671132,61903309 and 12271418)+2 种基金the National Key Research and Development Program of China (2020YFA0714200)Sichuan Science and Technology Program (2023NSFSC1355)the Applied Economics of Hunan Province.
文摘Long memory is an important phenomenon that arises sometimes in the analysis of time series or spatial data.Most of the definitions concerning the long memory of a stationary process are based on the second-order properties of the process.The mutual information between the past and future I_(p−f) of a stationary process represents the information stored in the history of the process which can be used to predict the future.We suggest that a stationary process can be referred to as long memory if its I_(p−f) is infinite.For a stationary process with finite block entropy,I_(p−f) is equal to the excess entropy,which is the summation of redundancies that relate the convergence rate of the conditional(differential)entropy to the entropy rate.Since the definitions of the I_(p−f) and the excess entropy of a stationary process require a very weak moment condition on the distribution of the process,it can be applied to processes whose distributions are without a bounded second moment.A significant property of I_(p−f) is that it is invariant under one-to-one transformation;this enables us to know the I_(p−f) of a stationary process from other processes.For a stationary Gaussian process,the long memory in the sense of mutual information is more strict than that in the sense of covariance.We demonstrate that the I_(p−f) of fractional Gaussian noise is infinite if and only if the Hurst parameter is H∈(1/2,1).
文摘A Lorenz map f : I --> I is a one dimensional piecewise monotone map with a single discontinuity c. Let [GRAPHICS] be the collection of all the preimsges of c. Authors prove that if C'(f) is countable then there exists M such that Card(omega(x)) less than or equal to M for all x is an element of I. If C'(f) is uncountable then omega(x) is uncountable for some x is an element of I. So f is asymptotically periodic if and only if C'(f) is countable.
基金supported by the National Key Re-search and Development Program of China(2020YFA0714200)supported by the Excellent Dissertation Cultivation Funds of Wuhan University of Technology(2018-YS-077)。
文摘We consider the topological behaviors of continuous maps with one topological attractor on compact metric space X.This kind of map is a generalization of maps such as topologically expansive Lorenz map,unimodal map without homtervals and so on.Under the finiteness and basin conditions,we provide a leveled A-R pair decomposition for such maps,and characterize α-limit set of each point.Based on weak Morse decomposition of X,we construct a bounded Lyapunov function V(x),which gives a clear description of orbit behavior of each point in X except a meager set.
