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11
S.,Limit theorems for additive functionals of a Markov Chain, version 1
, 2008
"... Abstract. Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the state space of the chain, with αtails with respect to π, α ∈ (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N 1/α ∑ N n Ψ(Xn) to an αstabl ..."
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Cited by 19 (7 self)
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Abstract. Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the state space of the chain, with αtails with respect to π, α ∈ (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N 1/α ∑ N n Ψ(Xn) to an αstable law. A “martingale approximation ” approach and “coupling ” approach give two different sets of conditions. We extend these results to continuous time Markov jump processes Xt, whose skeleton chain satisfies our assumptions. If waiting times between jumps have finite expectation, we prove convergence of N −1/α ∫ Nt 0 V (Xs)ds to a stable process. The result is applied to show that an appropriately scaled limit of solutions of a linear Boltzman equation is a solution of the fractional diffusion equation. 1.
CONVERGENCE OF A KINETIC EQUATION TO A FRACTIONAL DIFFUSION EQUATION
, 909
"... ABSTRACT. A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t), Y (t)) on (T × R), where T is the onedimensional torus. K(t) is a autonomous reversible jump process, with waiting times between two jumps with finite expectation valu ..."
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Cited by 3 (2 self)
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ABSTRACT. A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t), Y (t)) on (T × R), where T is the onedimensional torus. K(t) is a autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. v(K(s))ds, where v  ∼ 1 for small k. We prove that the rescaled process N −2/3Y (Nt) converge in distribution to a symmetric Lévy process, stable with index α = 3/2. Y (t) is an additive functional of K, defined as ∫ t 0 1. INTRODUCTION. The understanding of thermal conductance in both classical and quantum mechanical systems is one of the fundamental problems of nonequilibrium statistical mechanics. A particular aspect that has attracted much interest is the observation that autonomous translation invariant systems in dimensions one and two exhibit
Law of the Iterated Logarithm for stationary processes
, 2006
"... There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes · · · X−1,X0,X1, · · · whose partial sums Sn = X1 + · · · + Xn are of the form Sn = Mn + Rn, where Mn is a square integrable martingale with stationary increments and Rn is a ..."
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Cited by 2 (1 self)
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There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes · · · X−1,X0,X1, · · · whose partial sums Sn = X1 + · · · + Xn are of the form Sn = Mn + Rn, where Mn is a square integrable martingale with stationary increments and Rn is a remainder term for which E(R2 n) = o(n). Here we explore the Law of the Iterated Logarithm (LIL) for the same class of processes. Letting ‖ · ‖ denote the norm in L2 (P), a sufficient condition for the partial sums of a stationary process to have the form Sn = Mn + Rn is that n−3 2 ‖E(SnX0,X−1, · · ·) ‖ be summable. A sufficient condition for the LIL is only slightly stronger, requiring n−3 2 log 3 2(n)‖E(SnX0,X−1, · · ·) ‖ to be summable. As a byproduct of our main result, we obtain an improved statement of the Conditional Central Limit Theorem. Invariance principles are obtained as well. Key Word and Phrases: conditional central limit theorem, ergodic theorem, Fourier series, martingales, Markov chains, operators on L 2
LONG RANGE DEPENDENCE
"... Abstract. The notion of long range dependence is discussed from a variety of points of view, and a new approach is suggested. A number of related topics is also discussed, including connections with nonstationary processes, with ergodic theory, selfsimilar processes and fractionally differenced pr ..."
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Cited by 1 (1 self)
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Abstract. The notion of long range dependence is discussed from a variety of points of view, and a new approach is suggested. A number of related topics is also discussed, including connections with nonstationary processes, with ergodic theory, selfsimilar processes and fractionally differenced processes, heavy tails and light tails, limit theorems and large deviations. 1.
Asymptotic Properties of Maximum (Composite) Likelihood Estimators for Partially Ordered Markov Models
"... Partially ordered Markov models (POMMs) are Markov random fields (MRFs) with neighborhood structures derivable from an associated partially ordered set. The most attractive feature of POMMs is that their joint distributions can be written in closed and product form. Therefore, simulation and maximum ..."
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Cited by 1 (0 self)
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Partially ordered Markov models (POMMs) are Markov random fields (MRFs) with neighborhood structures derivable from an associated partially ordered set. The most attractive feature of POMMs is that their joint distributions can be written in closed and product form. Therefore, simulation and maximum likelihood estimation for the models is quite straightforward, which is not the case in general for MRF models. In practice, one often has to modify the likelihood to account for edge components; the resulting composite likelihood for POMMs is similarly straightforward to maximize. In this article, we use a martingale approach to derive the asymptotic properties of maximum (composite) likelihood estimators for POMMs. One of our results establishes that, under regularity conditions that are fairly easy to check and Dobrushin's condition for spatial mixing, the maximum composite likelihood estimator is consistent, asymptotically normal, and also asymptotically efficient. Key words and phrases...
NONMARKOVIAN LIMITS OF ADDITIVE FUNCTIONALS OF MARKOV PROCESSES
, 905
"... Abstract. In this paper we consider an additive functional of an observable V (x) of a Markov jump process. We assume that the law of the expected jump time t(x) under the invariant probability measure π of the skeleton chain belongs to the domain of attraction of a subordinator. Then, the scaled li ..."
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Abstract. In this paper we consider an additive functional of an observable V (x) of a Markov jump process. We assume that the law of the expected jump time t(x) under the invariant probability measure π of the skeleton chain belongs to the domain of attraction of a subordinator. Then, the scaled limit of the functional is a MittagLeffler proces, provided that Ψ(x): = V (x)t(x) is square integrable w.r.t. π. When the law of Ψ(x) belongs to a domain of attraction of a stable law the resulting process can be described by a composition of a stable process and the inverse of a subordinator and these processes are not necessarily independent. On the other hand when the singularities of Ψ(x) and t(x) do not overlap with large probability the law of the resulting process has some scaling invariance property. We provide an application of the results to a process that arises in quantum transport theory. 1.
FROM A KINETIC EQUATION TO A DIFFUSION UNDER AN ANOMALOUS SCALING
"... Abstract. A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t),i(t),Y(t)) on (T2 ×{1,2}×R 2), where T2 is the twodimensional torus. Here (K(t),i(t)) is an autonomous reversible jump process,with waitingtimesbetweentwojumpswith finit ..."
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Cited by 1 (1 self)
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Abstract. A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t),i(t),Y(t)) on (T2 ×{1,2}×R 2), where T2 is the twodimensional torus. Here (K(t),i(t)) is an autonomous reversible jump process,with waitingtimesbetweentwojumpswith finiteexpectation value but infinite variance. Y(t) is an additive functional v(K(s))ds, where v  ∼ 1 for small k. We prove that the rescaled process (N lnN) −1/2Y(Nt) converges in distribution to a twodimensional Brownian motion. As a consequence, the appropriatelyrescaledsolution of the Boltzmann equation converges to a diffusion equation. of K, defined as ∫ t 0 1.
CONVERGENCE TO LÉVY STABLE PROCESSES UNDER STRONG MIXING CONDITIONS
, 907
"... Abstract. For a strictly stationary sequence of random vectors in R d we study convergence of partial sums processes to Lévy stable process in the Skorohod space with J1topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationa ..."
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Abstract. For a strictly stationary sequence of random vectors in R d we study convergence of partial sums processes to Lévy stable process in the Skorohod space with J1topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing. 1.
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"... Convergence of clock process in random environments and aging in Bouchaud’s asymmetric trap model on the complete graph ..."
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Convergence of clock process in random environments and aging in Bouchaud’s asymmetric trap model on the complete graph