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140
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 52 (12 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.
Kernel dimension reduction in regression
, 2006
"... Acknowledgements. The authors thank the editor and anonymous referees for their helpful comments. The authors also thank Dr. Yoichi Nishiyama for his helpful comments on the uniform convergence of empirical processes. We would like to acknowledge support from JSPS KAKENHI 15700241, ..."
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Cited by 46 (16 self)
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Acknowledgements. The authors thank the editor and anonymous referees for their helpful comments. The authors also thank Dr. Yoichi Nishiyama for his helpful comments on the uniform convergence of empirical processes. We would like to acknowledge support from JSPS KAKENHI 15700241,
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 42 (6 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
Optimal control of spatially distributed systems
 IEEE Tran. on Automatic Control, September 2006, accepted. [Online]. Available: http://www.grasp.upenn.edu/ ∼ motee/ TACMoteeJ06SD.pdf
"... Abstract — In this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant dependent coupling functions over ar ..."
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Cited by 38 (5 self)
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Abstract — In this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant dependent coupling functions over arbitrary graphs. The key idea of the paper is the introduction of a special class of operators called spatially decaying (SD) operators. We study the structural properties of infinitehorizon linear quadratic optimal controllers for such systems by analyzing the spatial structure of the solution to the corresponding operator Lyapunov and Riccati equations. We prove that the kernel of the optimal feedback of each subsystem decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the system. I.
Universal kernels
 J. Machine Learning Research
, 2006
"... In this paper we investigate conditions on the features of a continuous kernel so that it may approximate an arbitrary continuous target function uniformly on any compact subset of the input space. A number of concrete examples are given of kernels with this universal approximating property. ..."
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Cited by 36 (3 self)
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In this paper we investigate conditions on the features of a continuous kernel so that it may approximate an arbitrary continuous target function uniformly on any compact subset of the input space. A number of concrete examples are given of kernels with this universal approximating property.
Statistical consistency of kernel canonical correlation analysis
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2007
"... While kernel canonical correlation analysis (CCA) has been applied in many contexts, the convergence of finite sample estimates of the associated functions to their population counterparts has not yet been established. This paper gives a mathematical proof of the statistical convergence of kernel CC ..."
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Cited by 25 (12 self)
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While kernel canonical correlation analysis (CCA) has been applied in many contexts, the convergence of finite sample estimates of the associated functions to their population counterparts has not yet been established. This paper gives a mathematical proof of the statistical convergence of kernel CCA, providing a theoretical justification for the method. The proof uses covariance operators defined on reproducing kernel Hilbert spaces, and analyzes the convergence of their empirical estimates of finite rank to their population counterparts, which can have infinite rank. The result also gives a sufficient condition for convergence on the regularization coefficient involved in kernel CCA: this should decrease as n −1/3, where n is the number of data.
On the orbital (in)stability of spatially periodic stationary solutions of generalized Kortewegde Vries equations
, 2010
"... In this paper we generalize previous work on the spectral and orbital stability of waves for infinitedimensional Hamiltonian systems to include those cases for which the skewsymmetric operator J is singular. We assume that J restricted to the orthogonal complement of its kernel has a bounded inv ..."
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Cited by 23 (7 self)
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In this paper we generalize previous work on the spectral and orbital stability of waves for infinitedimensional Hamiltonian systems to include those cases for which the skewsymmetric operator J is singular. We assume that J restricted to the orthogonal complement of its kernel has a bounded inverse. With this assumption and some further genericity conditions we (a) derive an unstable eigenvalue count for the appropriate linearized operator, and (b) show that the spectral stability of the wave implies its orbital (nonlinear) stability, provided there are no purely imaginary eigenvalues with negative Krein signature. We use our theory to investigate the (in)stability of spatially periodic waves to the generalized KdV equation for various power nonlinearities when the perturbation has the same period as that of the wave. Solutions of the integrable modified KdV equation are studied analytically in detail, as well as solutions with small amplitudes for higherorder pure power nonlinearities.
A RATE OF CONVERGENCE RESULT FOR THE LARGEST EIGENVALUE OF COMPLEX WHITE Wishart Matrices
, 2006
"... It has been recently shown that if X is an n × N matrix whose entries are i.i.d. standard complex Gaussian and l1 is the largest eigenvalue of X ∗ X, there exist sequences mn,N and sn,N such that (l1 − mn,N)/sn,N converges in distribution to W2, the Tracy–Widom law appearing in the study of the Gaus ..."
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Cited by 20 (0 self)
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It has been recently shown that if X is an n × N matrix whose entries are i.i.d. standard complex Gaussian and l1 is the largest eigenvalue of X ∗ X, there exist sequences mn,N and sn,N such that (l1 − mn,N)/sn,N converges in distribution to W2, the Tracy–Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of W2 is denoted F2. In this paper we show that, under the assumption that n/N → γ ∈ (0, ∞), we can find a function M, continuous and nonincreasing, and sequences ˜µn,N and ˜σn,N such that, for all real s0, there exists an integer N(s0,γ)for which, if (n ∧ N) ≥ N(s0,γ), we have, with ln,N = (l1 −˜µn,N) / ˜σn,N, ∀ s ≥ s0 (n ∧ N) 2/3 P(ln,N ≤ s) − F2(s)≤M(s0) exp(−s). The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution W2 is a good approximation to the empirical distribution of ln,N in simulations, an important fact from the point of view of (e.g., statistical) applications.
Noncoherent Capacity of Underspread Fading Channels
, 2008
"... We derive bounds on the noncoherent capacity of widesense stationary uncorrelated scattering (WSSUS) channels that are selective both in time and frequency, and are underspread, i.e., the product of the channel’s delay spread and Doppler spread is small. For input signals that are peak constrained ..."
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Cited by 18 (2 self)
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We derive bounds on the noncoherent capacity of widesense stationary uncorrelated scattering (WSSUS) channels that are selective both in time and frequency, and are underspread, i.e., the product of the channel’s delay spread and Doppler spread is small. For input signals that are peak constrained in time and frequency, we obtain upper and lower bounds on capacity that are explicit in the channel’s scattering function, are accurate for a large range of bandwidth and allow to coarsely identify the capacityoptimal bandwidth as a function of the peak power and the channel’s scattering function. We also obtain a closedform expression for the firstorder Taylor series expansion of capacity in the limit of large bandwidth, and show that our bounds are tight in the wideband regime. For input signals that are peak constrained in time only (and, hence, allowed to be peaky in frequency), we provide upper and lower bounds on the infinitebandwidth capacity and find cases when the bounds coincide and the infinitebandwidth capacity is characterized exactly. Our lower bound is closely related to a result by Viterbi (1967). The analysis in this paper is based on a discretetime discretefrequency approximation of WSSUS time and frequencyselective channels. This discretization explicitly takes into account the underspread