Limit Theorems in Hidden Markov Models

In this paper, under mild assumptions, we derive a law of large numbers, a central limit theorem with an error estimate, an almost sure invariance principle, and a variant of the Chernoff bound in finite-state hidden Markov models. These limit theorems are of interest in certain areas of information theory and statistics. Particularly, we apply the limit theorems to derive the rate of convergence of the maximum likelihood estimator in finite-state hidden Markov models.

[1]  Zhengyan Lin,et al.  Limit Theory for Mixing Dependent Random Variables , 1997 .

[2]  L. Arnold,et al.  Evolutionary Formalism for Products of Positive Random Matrices , 1994 .

[3]  R. Douc,et al.  CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR FOR GENERAL HIDDEN MARKOV MODELS , 2009, 0912.4480.

[4]  E. Seneta.  Non-negative Matrices and Markov Chains , 2008 .

[5]  Eytan Domany,et al.  The Entropy of a Binary Hidden Markov Process , 2005, ArXiv.

[6]  Yuval Peres,et al.  A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes , 2009, ArXiv.

[7]  Guangyue Han.  Limit theorems for the sample entropy of hidden Markov chains , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[8]  Wei Zeng,et al.  Simulation-Based Computation of Information Rates for Channels With Memory , 2006, IEEE Transactions on Information Theory.

[9]  Sandro Vaienti,et al.  FLUCTUATIONS OF THE METRIC ENTROPY FOR MIXING MEASURES , 2004 .

[10]  Tsachy Weissman,et al.  Entropy of Hidden Markov Processes and Connections to Dynamical Systems: Papers from the Banff International Research Station Workshop , 2011 .

[11]  Henry D. Pfister,et al.  The Capacity of Finite-State Channels in the High-Noise Regime , 2010, ArXiv.

[12]  R. Douc,et al.  Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime , 2004, math/0503681.

[13]  Richard C. Bradley,et al.  Introduction to strong mixing conditions , 2007 .

[14]  R. Douc,et al.  Asymptotics of the maximum likelihood estimator for general hidden Markov models , 2001 .

[15]  I. Ibragimov,et al.  Some Limit Theorems for Stationary Processes , 1962 .

[16]  Jun Luo,et al.  On the Entropy Rate of Hidden Markov Processes Observed Through Arbitrary Memoryless Channels , 2009, IEEE Transactions on Information Theory.

[17]  Hans-Andrea Loeliger,et al.  A Generalization of the Blahut–Arimoto Algorithm to Finite-State Channels , 2008, IEEE Transactions on Information Theory.

[18]  Hans-Andrea Loeliger,et al.  On the information rate of binary-input channels with memory , 2001, ICC 2001. IEEE International Conference on Communications. Conference Record (Cat. No.01CH37240).

[19]  T. Rydén.  Consistent and Asymptotically Normal Parameter Estimates for Hidden Markov Models , 1994 .

[20]  Brian H. Marcus,et al.  Asymptotics of Input-Constrained Binary Symmetric Channel Capacity , 2008, ArXiv.

[21]  V. V. Petrov.  Limit Theorems of Probability Theory: Sequences of Independent Random Variables , 1995 .

[22]  L. Gerencsér,et al.  Recursive estimation of Hidden Markov Models , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[23]  A. C. Berry.  The accuracy of the Gaussian approximation to the sum of independent variates , 1941 .

[24]  R. C. Bradley.  Basic properties of strong mixing conditions. A survey and some open questions , 2005, math/0511078.

[25]  Valerie Isham,et al.  Non‐Negative Matrices and Markov Chains , 1983 .

[26]  L. Baum,et al.  Statistical Inference for Probabilistic Functions of Finite State Markov Chains , 1966 .

[27]  N. Haydn,et al.  The Central Limit Theorem for uniformly strong mixing measures , 2009, 0903.1325.

[28]  En-Hui Yang,et al.  Non-asymptotic equipartition properties for independent and identically distributed sources , 2012, 2012 Information Theory and Applications Workshop.

[29]  V. V. Petrov.  On a Relation Between an Estimate of the Remainder in the Central Limit Theorem and the Law of the Iterated Logarithm , 1966 .

[30]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[31]  Andrea J. Goldsmith,et al.  Capacity of Finite State Channels Based on Lyapunov Exponents of Random Matrices , 2006, IEEE Transactions on Information Theory.

[32]  B. Leroux.  Maximum-likelihood estimation for hidden Markov models , 1992 .

[33]  Eytan Domany,et al.  Taylor series expansions for the entropy rate of Hidden Markov Processes , 2005, 2006 IEEE International Conference on Communications.

[34]  Thomas M. Cover,et al.  Elements of information theory (2. ed.) , 2006 .

[35]  Yuval Peres,et al.  Entropy Rate for Hidden Markov Chains with rare transitions , 2010, ArXiv.

[36]  P. Bickel,et al.  Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models , 1998 .

[37]  Tsachy Weissman,et al.  Asymptotic filtering and entropy rate of a hidden Markov process in the rare transitions regime , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[38]  Paul H. Siegel,et al.  On the achievable information rates of finite state ISI channels , 2001, GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270).

[39]  Brian H. Marcus,et al.  Derivatives of Entropy Rate in Special Families of Hidden Markov Chains , 2007, IEEE Transactions on Information Theory.

[40]  Brian H. Marcus,et al.  Asymptotics of Entropy Rate in Special Families of Hidden Markov Chains , 2010, IEEE Transactions on Information Theory.

[41]  Ioannis Kontoyiannis,et al.  Asymptotic Recurrence and Waiting Times for Stationary Processes , 1998 .

[42]  Thomas M. Cover,et al.  Elements of Information Theory: Cover/Elements of Information Theory, Second Edition , 2005 .

[43]  Brian H. Marcus,et al.  Entropy rate of continuous-state hidden Markov chains , 2010, 2010 IEEE International Symposium on Information Theory.

[44]  W. Philipp,et al.  Almost sure invariance principles for partial sums of weakly dependent random variables , 1975 .

[45]  Laurent Mevel,et al.  Asymptotical statistics of misspecified hidden Markov models , 2004, IEEE Transactions on Automatic Control.

[46]  Venkat Anantharam,et al.  An upper bound for the largest Lyapunov exponent of a Markovian product of nonnegative matrices , 2005, Theor. Comput. Sci..

[47]  Brian H. Marcus,et al.  Concavity of mutual information rate for input-restricted finite-state memoryless channels at high SNR , 2009, 2009 IEEE International Symposium on Information Theory.

[48]  Neri Merhav,et al.  Hidden Markov processes , 2002, IEEE Trans. Inf. Theory.

[49]  Peter J. Bickel,et al.  Inference in hidden Markov models I: Local asymptotic normality in the stationary case , 1996 .

[50]  Brian H. Marcus,et al.  Analyticity of Entropy Rate of Hidden Markov Chains , 2005, IEEE Transactions on Information Theory.

[51]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[52]  M. Loève.  On Almost Sure Convergence , 1951 .

[53]  S. Bernstein.  Sur l'extension du théoréme limite du calcul des probabilités aux sommes de quantités dépendantes , 1927 .

[54]  M. Kh. Reznik.  The Law of the Iterated Logarithm for Some Classes of Stationary Processes , 1968 .

[55]  J. Norris.  Appendix: probability and measure , 1997 .

[56]  Edgardo Ugalde,et al.  ON THE PRESERVATION OF GIBBSIANNESS UNDER SYMBOL AMALGAMATION , 2009, 0907.0528.

[57]  E. Seneta.  Non-negative Matrices and Markov Chains (Springer Series in Statistics) , 1981 .

[58]  Vladimir B. Balakirsky,et al.  On the entropy rate of a hidden Markov model , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[59]  John J. Birch.  Approximations for the Entropy for Functions of Markov Chains , 1962 .

[60]  V. Sharma,et al.  Entropy and channel capacity in the regenerative setup with applications to Markov channels , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[61]  Laurent Mevel,et al.  Exponential Forgetting and Geometric Ergodicity in Hidden Markov Models , 2000, Math. Control. Signals Syst..

[62]  Paul H. Siegel,et al.  On the capacity of finite state channels and the analysis of convolutional accumulate-m codes , 2003 .