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13
Exponential Stability for Nonlinear Filtering
, 1996
"... We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficie ..."
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Cited by 58 (2 self)
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We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discretecountable state spaces. R'esum'e Nous 'etudions la stabilit'e du filtre optimal par raport `a ses conditions initiales. Le taux de d'ecroissance exponentielle est calcul'e dans un cadre g'en'eral, pour temps discret et temps continu, en terme du coefficient de contraction de Birkhoff. Des crit`eres de stabilit'e exponentielle et des bornes explicites sur le taux sont calcul'ees pour les cas particuliers d'une diffusion sur une vari'ete compacte, ainsi que pour des chaines de Markov sur ...
On Certain Large Random Hermitian Jacobi Matrices with Applications to Wireless Communications
, 2007
"... In this paper we study the spectrum of certain large random Hermitian Jacobi matrices. These matrices are known to describe certain communication setups. In particular we are interested in an uplink cellular channel which models mobile users experiencing a softhandoff situation under joint multicel ..."
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Cited by 7 (3 self)
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In this paper we study the spectrum of certain large random Hermitian Jacobi matrices. These matrices are known to describe certain communication setups. In particular we are interested in an uplink cellular channel which models mobile users experiencing a softhandoff situation under joint multicell decoding. Considering rather general fading statistics we provide a closed form expression for the percell sumrate of this channel in highSNR, when an intracell TDMA protocol is employed. Since the matrices of interest are tridiagonal, their eigenvectors can be considered as sequences with second order linear recurrence. Therefore, the problem is reduced to the study of the exponential growth of products of two by two matrices. For the case where K users are simultaneously active in each cell, we obtain a series of lower and upper bound on the highSNR power offset of the percell sumrate, which are considerably tighter than previously known bounds. I.
Uniform markov renewal theory and ruin probabilities in Markov random
, 2004
"... Let {Xn,n ≥ 0} be a Markov chain on a general state space X with transition probability P and stationary probability π. Suppose an additive component Sn takes values in the real line R and is adjoined to the chain such that {(Xn,Sn),n ≥ 0} is a Markov random walk. In this paper, we prove a uniform M ..."
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Cited by 6 (2 self)
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Let {Xn,n ≥ 0} be a Markov chain on a general state space X with transition probability P and stationary probability π. Suppose an additive component Sn takes values in the real line R and is adjoined to the chain such that {(Xn,Sn),n ≥ 0} is a Markov random walk. In this paper, we prove a uniform Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(Xn,Sn),n ≥ 0}. To be more precise, for given b ≥ 0, define the stopping time τ = τ(b) = inf{n:Sn> b}. When a drift µ of the random walk Sn is 0, we derive a oneterm Edgeworth type asymptotic expansion for the first passage probabilities Pπ{τ < m} and Pπ{τ < m,Sm < c}, where m ≤ ∞, c ≤ b and Pπ denotes the probability under the initial distribution π. When µ ̸ = 0, Brownian approximations for the first passage probabilities with correction terms are derived. Applications to sequential estimation and truncated tests in random coefficient models and first passage times in products of random matrices are also given. 1. Introduction. Let {Xn,n ≥ 0
Surface Stretching For Ornstein Uhlenbeck Velocity Fields
, 1997
"... The present note deals with large time properties of the Lagrangian trajectories of a turbulent flow in IR 2 and IR 3 . We assume that the flow is driven by an incompressible timedependent random velocity field with Gaussian statistics. We also assume that the field is homogeneous in space and ..."
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Cited by 2 (0 self)
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The present note deals with large time properties of the Lagrangian trajectories of a turbulent flow in IR 2 and IR 3 . We assume that the flow is driven by an incompressible timedependent random velocity field with Gaussian statistics. We also assume that the field is homogeneous in space and stationary and Markovian in time. Such velocity fields can be viewed as (possibly infinite dimensional) OrnsteinUhlenbeck processes. In d spatial dimensions we established the (strict) positivity of the sum of the largest d \Gamma 1 Lyapunov exponents. As a consequences of this result, we prove the exponential stretching of surface areas (when d = 3) and of curve lengths (when d = 2) which confirms conjectures found in the theory of turbulent flows. 1 Introduction The mathematical model which we consider is commonly used for the time evolution of a collection of light pollutants carried by a turbulent flow. These pollutants do not affect the flow and can be viewed as passive tracers. To tr...
"Random" Random Matrix Products
"... . The paper deals with compositions of independent random bundle maps whose distributions form a stationary process which leads to study of Markov processes in random environments. A particular case of this situation is a product of independent random matrices with stationarily changing distribution ..."
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Cited by 1 (1 self)
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. The paper deals with compositions of independent random bundle maps whose distributions form a stationary process which leads to study of Markov processes in random environments. A particular case of this situation is a product of independent random matrices with stationarily changing distributions. I obtain results concerning invariant ltrations for such systems, positivity and simplicity of the largest Lyapunov exponent, as well as the central limit theorem type results. An application to random harmonic functions and measures is also considered. Continuous time versions of these results are also discussed which yield applications to linear stochastic dierential equations in random environments. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). Typeset by A M ST E X 1 2 Y. KIFER 1. Introduction Starting from the beginning of sixties a lot of work has been done on products o...
Positive KolmogorovSinai entropy for the Standard map
, 1999
"... We prove that the KolmogorovSinai entropy of the ChirikovStandard map Tf : (x; y) 7! (2x \Gamma y + f(x); x) with f(x) = sin(x) with respect to the invariant Lebesgue measure on the twodimensional torus is bounded below by log(=2) \Gamma C() with C() = arcsinh(1=) + log(2= p 3). For ? 0 = (8=( ..."
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We prove that the KolmogorovSinai entropy of the ChirikovStandard map Tf : (x; y) 7! (2x \Gamma y + f(x); x) with f(x) = sin(x) with respect to the invariant Lebesgue measure on the twodimensional torus is bounded below by log(=2) \Gamma C() with C() = arcsinh(1=) + log(2= p 3). For ? 0 = (8=(6 \Gamma 3 p 3)) 1=2 = 3:1547:::, the entropy of T sin is positive. This result is stable in Banach spaces of realanalytic symplectic maps: each ChirikovStandard map with ? 0 is contained in an open set of realanalytic, in general nonergodic areapreserving diffeomorphisms with positive entropy. Moreover, there is a C 0 dense set of realanalytic maps f in C 0 (T) for which the entropy of Tf is positive. The Lyapunov exponent estimates hold for a fixed cocycle uniformly for the entire group of measure preserving maps on the torus. This establishes new families of discrete ergodic onedimensional Schrodinger operators (Lu)n = un+1 + un\Gamma1 + cos(xn)un with no absolutely contin...
Sequential detection of Markov targets with trajectory estimation
, 2008
"... The problem of detection and possible estimation of a signal generated by a dynamic system when a variable number of noisy measurements can be taken is here considered. Assuming a Markov evolution of the system (in particular, the pair signalobservation forms a hidden Markov model), a sequential pr ..."
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The problem of detection and possible estimation of a signal generated by a dynamic system when a variable number of noisy measurements can be taken is here considered. Assuming a Markov evolution of the system (in particular, the pair signalobservation forms a hidden Markov model), a sequential procedure is proposed, wherein the detection part is a sequential probability ratio test (SPRT) and the estimation part relies upon a maximumaposteriori (MAP) criterion, gated by the detection stage (the parameter to be estimated is the trajectory of the state evolution of the system itself). A thorough analysis of the asymptotic behaviour of the test in this new scenario is given, and sufficient conditions for its asymptotic optimality are stated, i.e. for almost sure minimization of the stopping time and for (firstorder) minimization of any moment of its distribution. An application to radar surveillance problems is also examined.
An abstract continuity theorem for Lyapunov exponents of linear cocycles and an application to random cocycles, preprint
, 2014
"... Abstract. We devise an abstract scheme to prove continuity of the Lyapunov exponents for a general class of linear cocycles. The main assumption is a large deviation type (LDT) estimate, uniform in the data. We provide a modulus of continuity that depends explicitly on the sharpness of the LDT estim ..."
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Abstract. We devise an abstract scheme to prove continuity of the Lyapunov exponents for a general class of linear cocycles. The main assumption is a large deviation type (LDT) estimate, uniform in the data. We provide a modulus of continuity that depends explicitly on the sharpness of the LDT estimate. Moreover, we derive such estimates for a class of irreducible random cocycles of Markovian type, thus proving Hölder continuity of their Lyapunov exponents. This is a preliminary version. 1.
Exponential decay rate of the filter’s dependence
, 2009
"... on the initial distribution ..."
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1On Certain Large Random Hermitian Jacobi Matrices with Applications to Wireless Communications
"... In this paper we study the spectrum of certain large random Hermitian Jacobi matrices. These matrices are known to describe certain communication setups. In particular we are interested in an uplink cellular channel which models mobile users experiencing a softhandoff situation under joint multicel ..."
Abstract
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In this paper we study the spectrum of certain large random Hermitian Jacobi matrices. These matrices are known to describe certain communication setups. In particular we are interested in an uplink cellular channel which models mobile users experiencing a softhandoff situation under joint multicell decoding. Considering rather general fading statistics we provide a closed form expression for the percell sumrate of this channel in highSNR, when an intracell TDMA protocol is employed. Since the matrices of interest are tridiagonal, their eigenvectors can be considered as sequences with second order linear recurrence. Therefore, the problem is reduced to the study of the exponential growth of products of two by two matrices. For the case where users are simultaneously active in each cell, we obtain a series of lower and upper bound on the highSNR power offset of the percell sumrate, which are considerably tighter than previously known bounds. I.