A tree is a graph g t, e that is connected and contains no circuits. Toss the coin several times to create x 1,x 2, etc. It states that tropical configurations can always be approximated by homogeneous configurations. The asymptotic equipartition property for nonhomogeneous. Quotients of markov chains and asymptotic properties of the stationary distribution of the markov chain associated to an evolutionary algorithm boris mitavskiy. Asymptotic equipartition property, which states the outcomes of a stochastic process can be divided into two groups. Strong law of large numbers and asymptotic equipartition. T be an asymptotic evenodd markov chain indexed by a homogeneous tree defined by definition 2. Key words cayley tree, nonsymmetric markov chain fields, strong law of large numbers, asymptotic equipartition property 2000 mr subject classification 60j25, 60f15 1 introduction we begin with notations and definitions, most of which can be found in l3. These nonasymptotic equipartition properties are instrumental to the development of nonasymptotic coding including both source and channel coding results in information theory in the same way as the asymptotic equipartition property to all asymptotic coding theorems established so far in information theory.
Improving the asymptotic performance of markov chain. In this article, we study the asymptotic equipartition property aep for asymptotic circular markov chains. Improving the asymptotic performance of markov chain monte. Journal of jiangsu universitynatural science eidt, 2012, 336. The asymptotic equipartition property for asymptotic circular. Nonnegativity implies that this martingale converges. We also proved that typical encodings yield the asymptotic equipartition property. Then we prove the generalized strong laws of large numbers and the generalized asymptotic equipartition property for delayed sums of finite nonhomogeneous markov chains indexed by a homogeneous tree. Information theory winter 2017 lecture 4 jan 19 lecturer. The asymptotic equipartition property is a basic theorem in information theory. Markov chain is time invariant, with transition probability matrix p if. Types of convergence weak law of large numbers strong. Asymptotic equipartition property law of large number for product of random variables aep and consequences dr.
A maximal entropy stochastic process for a timed automaton. In this paper, we study the strong law of large numbers. Divergence rate of asymptotic circular markov chains. In addition, we show that the solutions to certain informationoptimization problems are lipschitzcon\tinuous with respect to the asymptotic kolmogorovsinai distance. It is a direct consequence of the weak law of large numbers. A maximal entropy stochastic process for a timed automaton nicolas basset to cite this version. For any two states, the first passage time probability in n steps is defined as follows and this probability is related to the ever reaching probability. For an irreducible and recurrent markov chain, every nonnegative harmonic function is a constant. University of illinois at chicago ece 534, natasha devroye chapter 3 outline strong vs. The asymptotic equipartition property is an instance of the weak law of large numbers. I hope to convince you that what the asymptotic equipartition property is, is simply the week law of large numbers applied to the log pmf.
In a markov chain, later nodes depend only on previous nodes. Types of convergence weak law of large numbers strong law of large numbers asymptotic equipartition property lecture outline stochastic processes markov chains entropy rate random walks on graphs. Research article strong law of large numbers for countable. First, the definition of an asymptotic circular markov chain is introduced. Strong law of large numbers asymptotic equipartition property lecture outline stochastic processes markov chains entropy rate random walks on graphs hidden markov models reading. The asymptotic equipartition property of markov chains in single. A maximal entropy stochastic process for a timed automaton nicolas basseta,b, aligm, university parisest marnelavall ee and cnrs, france bliafa, university paris diderot and cn. This is why shannons original argument was for markov chains. The asymptotic equipartition property in reinforcement. In this paper, we study the asymptotic equipartition property aep for m order nonhomogeneous markov information source. Manyofthe proofs will be rigorous, to display more generally useful techniques also for later chapters.
The asymptotic equipartition property in reinforcement learning and its relation to return maximization kazunori iwataa, kazushi ikedab, hideaki sakaib afaculty of information sciences, hiroshima city university, 341 ozukahigashi, asaminamiku, hiroshima 73194, japan. Strong laws of large numbers for asymptotic evenodd markov. Comparison of asymptotic variances of inhomogeneous. Strong laws of large numbers for asymptotic evenodd. We prove the strong law of large numbers and the asymptotic equipartition property aep for finite nonhomogeneous markov chains indexed by trees. The asymptotic equipartition property of markov chains in. Types of convergence weak law of large numbers strong law of large numbers asymptotic equipartition property lecture outline stochastic processes markov chains entropy rate random walks on graphs hidden markov models reading. For a regular markov chain, the stationary distribution is unique and the asymptotic distribution is the stationary distribution. Comparison of asymptotic variances of inhomogeneous markov. The asymptotic equipartition property in information theory, the analog of the law of large numbers is the asymptotic equipartition property aep. In this paper, we study the asymptotic equipartition property aep for a nonhomogeneous markov information source. Huang and yang have studied the strong law of large numbers and the asymptotic equipartition property aep for nite markov chains indexed by an in nite tree with uniformly bounded degree. Asymptotic properties of the mle in this part of the course, we will consider the asymptotic properties of the maximum likelihood estimator. Asymptotic equipartition property introductioni the law of large numbers states that for i.
Asymptotic equipartition property aep and entropy rates. Strong law of large numbers for countable markov chains. Divergence rate of asymptotic circular markov chainsj. But partly because i want to talk about something which is called the asymptotic equipartition property. In the case where the markov chain is nonreversible, i. Ye, the asymptotic equipartition property for nonhomogeneous markov chains indexed by a homogeneous tree, ieee transactions on information theory, vol. Penghua wang, april 2, 2012 information theory, chap. For markov chains with a large state space and lack of structure, simulation.
It is easy to check that hxn is a nonnegative martingale. In the proof, a new technique for the study of strong limit theorems of markov chains is extended to the case of markov chain fields. A simple consequence of this definition is that an i. Definition aperiodicity and irreducibility if xn is a markov chain taking values in x, with x m, then. In this paper, we study the strong law of large numbers of markov chains in. Some limit properties of random transition probability for. As corollaries, we get several limit theorems about frequency of occurrence of states and a limit theorem about the entropy density for m order. Nonasymptotic equipartition properties for independent.
Evaluating first passage times in markov chains 31 among the markov chain characteristics, the first passage times play an important role. In the following, we will obtain the asymptotic equipartition property for asymptotic evenodd markov chains indexed by a homogeneous tree which generalizes the results of. At first we study a generalized strong limit theorem for nonhomogeneous markov chains indexed by a homogeneous tree. The ideas and methods are generalized to a large class of averages and to vector valued markov chains. The there exists a code that maps sequences xn of length n into binary strings such that the mappings is onetoone and therefore invertible and e 1 n. One kind of analysis in this framework that has been very successful is asymptotic analysis of steadystate tail probabilities. And because of those long words you believe this has to be very complicated. In this lecture, we discuss the asymptotic equipartition property aep regarding the sequences output by a stochastic source. The asymptotic equipartition property holds if the process is white, in which case the time samples are i. We first give a limit theorem for the averages of the functions of two variables of this information source by using the convergence theorem for the martingale difference sequence. Markov property thatq is a geometric random variable with probability of success.
The asymptotic equipartition property for asymptotic. In this context we prove an asymptotic equipartition property for configurations. Entropy, markov chains, and huffman coding department of. As a consequence it has the asymptotic equipartition property and thus the random sampling w. In particular, we will study issues of consistency, asymptotic normality, and e. Strong law of large numbers and asymptotic equipartition property. In this paper, we study the strong law of large numbers of markov chains in singleinfinite markovian environment on countable state space. Information source mathematics markov information source. Some strong limit properties, such as the strong law of large numbers and the asymptotic equipartition property, for nonhomogeneous markov chains indexed by t, are established.
Markov chains, in particular, the markov chains of gim1 type and mgi1 type in neuts 17, 20. This structure facilitates detailed analysis of many models of interest. The asymptotic equipartition property is a basic theorem in informa tion theory. Types of convergence weak law of large numbers strong law of.
The asymptotic equipartition property for nonhomogeneous markov chains indexed by a homogeneous tree. Quotients of markov chains and asymptotic properties of the. Asymptotic equipartition property for markov chains17 4. A note on the strong law of large numbers for markov. Asymptotic properties of the markov chain model method of finding markov chains generators of doi. Quotients of markov chains and asymptotic properties of. Markov chains markov chains are discrete state space processes that have the markov property. A note on the strong law of large numbers for markov chains.
The asymptotic equipartition property in reinforcement learning and its relation to return maximization kazunori iwataa, kazushi ikedab, hideaki sakaib afaculty of information sciences, hiroshima city university, 341 ozukahigashi, asaminamiku, hiroshima 73194, japan bdepartment of systems science, graduate school of informatics, kyoto university, yoshida. On the transition diagram, x t corresponds to which box we are in at stept. Asymptotic properties of the markov chain model method of. Weak typicality convergence asymptotic equipartition property theorem highprobability sets and the typical set. Ieee transactions on information theory 2007, 53 9. Yao xie, ece587, information theory, duke university.
199 142 308 196 1219 156 1393 1158 1056 469 1549 427 559 1193 895 1018 138 628 1256 78 98 631 1067 348 1041 155 178 114 858 254 1243 1450