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139
Wavelet Threshold Estimators for Data With Correlated Noise
, 1994
"... Wavelet threshold estimators for data with stationary correlated noise are constructed by the following prescription. First, form the discrete wavelet transform of the data points. Next, apply a leveldependent soft threshold to the individual coefficients, allowing the thresholds to depend on the l ..."
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Cited by 182 (13 self)
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Wavelet threshold estimators for data with stationary correlated noise are constructed by the following prescription. First, form the discrete wavelet transform of the data points. Next, apply a leveldependent soft threshold to the individual coefficients, allowing the thresholds to depend on the level in the wavelet transform. Finally, transform back to obtain the estimate in the original domain. The threshold used at level j is s j p 2 log n, where s j is the standard deviation of the coefficients at that level, and n is the overall sample size. The minimax properties of the estimators are investigated by considering a general problem in multivariate normal decision theory, concerned with the estimation of the mean vector of a general multivariate normal distribution subject to squared error loss. An ideal risk is obtained by the use of an `oracle' that provides the optimum diagonal projection estimate. This `benchmark' risk can be considered in its own right as a measure of the s...
Stability, queue length and delay of deterministic and stochastic queueing networks
 IEEE Transactions on Automatic Control
, 1994
"... Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of ..."
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Cited by 171 (20 self)
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Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of a queue has an exponential tail with rate `. To answer these two types of stability problems, we introduce two new notions of traffic characterization: minimum envelope rate (MER) and minimum envelope rate with respect to `. Based on these two new notions of traffic characterization, we develop a set of rules for network operations such as superposition, inputoutput relation of a single queue, and routing. Specifically, we show that (i) the MER of a superposition process is less than or equal to the sum of the MER of each process, (ii) a queue is stable in the sense of bounded queue length if the MER of the input traffic is smaller than the capacity, (iii) the MER of a departure process from a stable queue is less than or equal to that of the input process (iv) the MER of a routed process from a departure process is less than or equal to the MER of the departure process multiplied by the MER of the routing process. Similar results hold for MER with respect to ` under a further assumption of independence. These rules provide a natural way to analyze feedforward networks with multiple classes of customers. For single class networks with nonfeedforward routing, we provide a new method to show that similar stability results hold for such networks under the FCFS policy. Moreover, when restricting to the family of twostate Markov modulated arrival processes, the notion of MER with respect to ` is shown to be
Multipleantenna channel hardening and its implications for rate feedback and scheduling
 IEEE Transactions on Information Theory
, 2004
"... Wireless data traffic is expected to grow over the next few years and the technologies that will provide data services are still being debated. One possibility is to use multiple antennas at basestations and terminals to get very high spectral efficiencies in rich scattering environments. Such multi ..."
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Cited by 106 (2 self)
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Wireless data traffic is expected to grow over the next few years and the technologies that will provide data services are still being debated. One possibility is to use multiple antennas at basestations and terminals to get very high spectral efficiencies in rich scattering environments. Such multipleinput multipleoutput (MIMO) channels can then be used in conjunction with scheduling and ratefeedback algorithms to further increase channel throughput. This paper provides an analysis of the expected gains due to scheduling and bits needed for rate feedback. Our analysis requires an accurate approximation of the distribution of the MIMO channel mutual information. Because the exact distribution of the mutual information in a Rayleigh fading environment is difficult to analyze, we prove a central limit theorem for MIMO channels with a large number of antennas. While the growth in average mutual information (capacity) of a MIMO channel with the number of antennas is well understood, it turns out that the variance of the mutual information can grow very slowly or even shrink as the number of antennas grows. We discuss implications of this “channelhardening ” result for data and voice services, scheduling and rate feedback. We also briefly discuss the implications when shadow fading effects are included. Index Terms—Wireless communications, transmit diversity, receive diversity, fading channels 1
The transport capacity of wireless networks over fading channels
 IEEE Transactions on Information Theory
, 2005
"... Abstract — We consider networks consisting of nodes with radios, and without any wired infrastructure, thus necessitating all communication to take place only over the shared wireless medium. The main focus of this paper is on the effect of fading in such wireless networks. We examine the attenuatio ..."
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Cited by 63 (3 self)
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Abstract — We consider networks consisting of nodes with radios, and without any wired infrastructure, thus necessitating all communication to take place only over the shared wireless medium. The main focus of this paper is on the effect of fading in such wireless networks. We examine the attenuation regime where either the medium is absorptive, a situation which generally prevails, or the path loss exponent is greater than 3. We study the transport capacity, defined as the supremum over the set of feasible rate vectors of the distance weighted sum of rates. We consider two assumption sets. Under the first assumption set, which essentially requires only a mild time average type of bound on the fading process, we show that the transport capacity can grow no faster than ¢¡¤£¦ ¥ , where £ denotes the number of nodes, even when the channel state information (CSI) is available noncausally at both the transmitters and the receivers. This assumption includes common models of stationary ergodic channels; constant, frequency selective channels; flat, rapidly varying channels; and flat slowly varying channels. In the second assumption set, which essentially features an independence, time average of expectation, and nonzeroness condition on the fading process, we constructively show how to achieve transport capacity of § even when the CSI is unknown to both the transmitters and the receivers, provided that every node has an appropriately nearby node. This assumption set includes common models of i.i.d. channels; constant, flat channels; and constant, frequency selective channels. The transport capacity is achieved by nodes only communicating with neighbors, and only using pointtopoint coding. The thrust of these results is that the multihop strategy, towards which much protocol development activity is currently targeted, is appropriate for fading environments. The low attenuation regime is open. Index Terms — Wireless networks, fading channels, capacity, transport capacity.
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros ..."
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Cited by 47 (11 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
Subexponential Asymptotics of a MarkovModulated Random Walk with Queueing Applications
, 1996
"... Let f(Xn; Jn)g be a stationary Markovmodulated random walk on R\Theta E (E finite), defined by its probability transition matrix measure F = fF ij g; F ij (B) = P[X 1 2 B; J 1 = jjJ 0 = i]; B 2 B(R); i; j 2 E. If F ij ([x; 1))=(1 \Gamma H(x)) ! W ij 2 [0; 1), as x! 1, for some longtailed distribut ..."
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Cited by 45 (15 self)
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Let f(Xn; Jn)g be a stationary Markovmodulated random walk on R\Theta E (E finite), defined by its probability transition matrix measure F = fF ij g; F ij (B) = P[X 1 2 B; J 1 = jjJ 0 = i]; B 2 B(R); i; j 2 E. If F ij ([x; 1))=(1 \Gamma H(x)) ! W ij 2 [0; 1), as x! 1, for some longtailed distribution function H, then the ascending ladder heights matrix distribution G+ (x) (right WienerHopf factor) has longtailed asymptotics. If EXn! 0, at least one W ij? 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by P \Theta sup n0 Sn? x
Convergence rates of posterior distributions
 Ann. Statist
, 2000
"... We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, D ..."
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Cited by 43 (11 self)
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We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, Dirichlet processes and interval censoring. 1. Introduction. Suppose
Moment Inequalities for Functions of Independent Random Variables
"... this paper is to provide such generalpurpose inequalities. Our approach is based on a generalization of Ledoux's entropy method (see [26, 28]). Ledoux's method relies on abstract functional inequalities known as logarithmic Sobolev inequalities and provide a powerful tool for deriving exponential i ..."
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Cited by 39 (9 self)
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this paper is to provide such generalpurpose inequalities. Our approach is based on a generalization of Ledoux's entropy method (see [26, 28]). Ledoux's method relies on abstract functional inequalities known as logarithmic Sobolev inequalities and provide a powerful tool for deriving exponential inequalities for functions of independent random variables, see Boucheron, Massart, and AMS 1991 subject classifications. Primary 60E15, 60C05, 28A35; Secondary 05C80 Key words and phrases. Moment inequalities, Concentration inequalities; Empirical processes; Random graphs Supported by EU Working Group RANDAPX, binational PROCOPE Grant 05923XL The work of the third author was supported by the Spanish Ministry of Science and Technology and FEDER, grant BMF200303324 Lugosi [6, 7], Bousquet [8], Devroye [14], Massart [30, 31], Rio [36] for various applications. To derive moment inequalities for general functions of independent random variables, we elaborate on the pioneering work of Latala and Oleszkiewicz [25] and describe socalled #Sobolev inequalities which interpolate between Poincare's inequality and logarithmic Sobolev inequalities (see also Beckner [4] and Bobkov's arguments in [26])
Minimizing regret with label efficient prediction
 IEEE Trans. Inform. Theory
, 2005
"... Abstract. We investigate label efficient prediction, a variant of the problem of prediction with expert advice, proposed by Helmbold and Panizza, in which the forecaster does not have access to the outcomes of the sequence to be predicted unless he asks for it, which he can do for a limited number o ..."
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Cited by 39 (6 self)
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Abstract. We investigate label efficient prediction, a variant of the problem of prediction with expert advice, proposed by Helmbold and Panizza, in which the forecaster does not have access to the outcomes of the sequence to be predicted unless he asks for it, which he can do for a limited number of times. We determine matching upper and lower bounds for the best possible excess error when the number of allowed queries is a constant. We also prove that a query rate of order (ln n)(ln ln n) 2 /n is sufficient for achieving Hannan consistency, a fundamental property in gametheoretic prediction models. Finally, we apply the label efficient framework to pattern classification and prove a label efficient mistake bound for a randomized variant of Littlestone’s zerothreshold Winnow algorithm. 1