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27
Exponential Forgetting and Geometric Ergodicity in Hidden Markov Models
 Math. Control Signals Systems
"... We consider a hidden Markov model with multidimensional observations, and with misspecification, i.e. the assumed coefficients (transition probability matrix, and observation conditional densities) are possibly different from the true coefficients. Under mild assumptions on the coefficients of both ..."
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Cited by 41 (2 self)
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We consider a hidden Markov model with multidimensional observations, and with misspecification, i.e. the assumed coefficients (transition probability matrix, and observation conditional densities) are possibly different from the true coefficients. Under mild assumptions on the coefficients of both the true and the assumed models, we prove that : (i) the prediction filter forgets almost surely its initial condition exponentially fast, and (ii) the extended Markov chain, whose components are : the unobserved Markov chain, the observation sequence, and the prediction filter, is geometrically ergodic, and has a unique invariant probability distribution. 1 Introduction Let fXn ; n 0g and fYn ; n 0g be two random sequences defined on the probability space(\Omega ; F ; P ffl ), with values in the finite set S = f1; \Delta \Delta \Delta ; Ng and in R d respectively. It is assumed that : ffl The unobserved state sequence fXn ; n 0g is a timehomogeneous Markov chain with transition p...
The capacity of channels with feedback
 IEEE Trans. Information Theory
, 2009
"... We introduce a general framework for treating channels with memory and feedback. First, we generalize Massey’s concept of directed information [23] and use it to characterize the feedback capacity of general channels. Second, we present coding results for Markov channels. This requires determining a ..."
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Cited by 38 (2 self)
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We introduce a general framework for treating channels with memory and feedback. First, we generalize Massey’s concept of directed information [23] and use it to characterize the feedback capacity of general channels. Second, we present coding results for Markov channels. This requires determining appropriate sufficient statistics at the encoder and decoder. Third, a dynamic programming framework for computing the capacity of Markov channels is presented. Fourth, it is shown that the average cost optimality equation (ACOE) can be viewed as an implicit singleletter characterization of the capacity. Fifth, scenarios
Asymptotic stability of the Wonham filter: ergodic and nonergodic signals
 SIAM J. Control Optim
"... Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the ..."
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Cited by 25 (13 self)
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Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the nonergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure. Key words. Nonlinear filtering, stability, Wonham filter
Processes with Long Memory: Regenerative Construction and Perfect
 k=1 j1,...,jk∈A j1∈A ( = ξt a + ∑ j1∈A αj1ξt−j1 αj1Xt−j1 ) αj1ξt−j1 · · · αjkξt−j1−···−jka ( a + +∞∑ k=2 ) . □ αj2ξt−j1−j2 . . . αjk ξt−j1−j2−···−jk
, 2002
"... We present a perfect simulation algorithm for stationary processes indexed by Z, with summable memory decay. Depending on the decay, we construct the process on finite or semiinfinite intervals, explicitly from an i.i.d. uniform sequence. Even though the process has infinite memory, its value at ti ..."
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Cited by 24 (2 self)
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We present a perfect simulation algorithm for stationary processes indexed by Z, with summable memory decay. Depending on the decay, we construct the process on finite or semiinfinite intervals, explicitly from an i.i.d. uniform sequence. Even though the process has infinite memory, its value at time 0 depends only on a finite, but random, number of these uniform variables. The algorithm is based on a recent regenerative construction of these measures by Ferrari, Maass, Martínez and Ney. As applications, we discuss the perfect simulation of binary autoregressions and Markov chains on the unit interval. 1. Introduction. In
Basic Properties of the Projective Product with Application to Products of ColumnAllowable Nonnegative Matrices
, 2000
"... We use basic properties of the projectire product to obtain exponential bounds for the Lipschitz constant associated with the projectire product of columnallowable nonnegative matrices. We obtain similar bounds for the associated linear tangent maps. ..."
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Cited by 18 (1 self)
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We use basic properties of the projectire product to obtain exponential bounds for the Lipschitz constant associated with the projectire product of columnallowable nonnegative matrices. We obtain similar bounds for the associated linear tangent maps.
Capacity, mutual information, and coding for finitestate Markov channels
 IEEE Trans. Inform. Theory
, 1996
"... Abstract The FiniteState Markov Channel (FSMC) is a discretetime varying channel whose variation is determined by a finitestate Markov process. These channels have memory due to the Markov channel variation. We obtain the FSMC capacity as a function of the conditional channel state probability. W ..."
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Cited by 15 (2 self)
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Abstract The FiniteState Markov Channel (FSMC) is a discretetime varying channel whose variation is determined by a finitestate Markov process. These channels have memory due to the Markov channel variation. We obtain the FSMC capacity as a function of the conditional channel state probability. We also show that for i.i.d. channel inputs, this conditional probability converges weakly, and the channel's mutual information is then a closedform continuous function of the input distribution. We next consider coding for FSMCs. In general, the complexity of maximumlikelihood decoding grows exponentially with the channel memory length. Therefore, in practice, interleaving and memoryless channel codes are used. This technique results in some performance loss relative to the inherent capacity of channels with memory. We propose a maximumlikelihood decisionfeedback decoder with complexity that is independent of the channel memory. We calculate the capacity and cutoff rate of our technique, and show that it preserves the capacity of certain FSMCs. We also compare the performance of the decisionfeedback decoder with that of interleaving and memoryless channel coding on a fading channel with 4PSK modulation.
Geometric Ergodicity in Hidden Markov Models
 PUBLICATION INTERNE 1028, IRISA
, 1996
"... We consider an hidden Markov model with multidimensional observations, and with misspecification, i.e. the assumed coefficients (transition probability matrix, and observation conditional densities) are possibly different from the true coefficients. Under mild assumptions on the coefficients of both ..."
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Cited by 15 (3 self)
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We consider an hidden Markov model with multidimensional observations, and with misspecification, i.e. the assumed coefficients (transition probability matrix, and observation conditional densities) are possibly different from the true coefficients. Under mild assumptions on the coefficients of both the true and the assumed models, we prove that : (i) the prediction filter, and its gradient w.r.t. some parameter in the model, forget almost surely their initial condition exponentially fast, and (ii) the extended Markov chain, whose components are : the unobserved Markov chain, the observation sequence, the prediction filter, and its gradient, is geometrically ergodic and has a unique invariant probability distribution.
The compound channel capacity of a class of finitestate channels
 IEEE Trans. Inform. Theory
, 1998
"... Abstract — A transmitter and receiver need to be designed to guarantee reliable communication on any channel belonging to a given family of finitestate channels defined over common finite input, output, and state alphabets. Both the transmitter and receiver are assumed to be ignorant of the channel ..."
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Cited by 14 (1 self)
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Abstract — A transmitter and receiver need to be designed to guarantee reliable communication on any channel belonging to a given family of finitestate channels defined over common finite input, output, and state alphabets. Both the transmitter and receiver are assumed to be ignorant of the channel over which transmission is carried out and also ignorant of its initial state. For this scenario we derive an expression for the highest achievable rate. As a special case we derive the compound channel capacity of a class of Gilbert–Elliott channels. Index Terms—Compound channel, error exponents, finitestate channels, Gilbert–Elliott channel, universal decoding. I.
R.Liptser, Stability of nonlinear filters in nonmixing
"... The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by, so called, mixing condition. The ..."
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Cited by 14 (6 self)
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The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is a uniform positiveness of this density. We show that this requirement might be weakened regardless of an observation process structure.
Design and performance of highspeed communication systems over timevarying radio channels
 ELEC. ENGIN. COMPUT. SCIENCE
, 1994
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