A lower bound in Nehari's theorem on the polydisc

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A performance lower bound for quadratic timing recovery accounting for the symbol transition density

The symbol transition density in a digitally modulated signal affects the performance of practical synchronization schemes designed for timing recovery. This paper focuses on the derivation of simple performance limits for the estimation of the time delay of a noisy linearly modulated signal in the presence of various degrees of symbol correlation produced by the varioustransition densities in the symbol streams. The paper develops high- and low-signal-to-noise ratio (SNR) approximations of the so-called (Gaussian) unconditional Cramér–Rao bound (UCRB),as well as general expressions that are applicable in all ranges of SNR. The derived bounds are valid only for the class of quadratic, non-data-aided (NDA) timing recovery schemes. To illustrate the validity of the derived bounds, they are compared with the actual performance achieved by some well-known quadratic NDA timing recovery schemes. The impact of the symbol transitiondensity on the classical threshold effect present in NDA timing recovery schemes is also analyzed. Previous work on performancebounds for timing recovery from various authors is generalized and unified in this contribution.

A lower bound in Nehari's theorem on the polydisc

By theorems of Ferguson and Lacey ($d=2$) and Lacey and Terwilleger ($d>2$), Nehari's theorem is known to hold on the polydisc $\D^d$ for $d>1$, i.e., if $H_\psi$ is a bounded Hankel form on $H^2(\D^d)$ with analytic symbol $\psi$, then there is a function $\varphi$ in $L^\infty(\T^d)$ such that $\psi$ is the Riesz projection of $\varphi$. A method proposed in Helson's last paper is used to show that the constant $C_d$ in the estimate $\|\varphi\|_\infty\le C_d \|H_\psi\|$ grows at least exponentially with $d$; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.

Minimax lower bounds for the two-armed bandit problem

We obtain minimax lower bounds on the regret for the classicaltwo--armed bandit problem. We provide a finite--sample minimax version of the well--known log $n$ asymptotic lower bound of Lai and Robbins. Also, in contrast to the log $n$ asymptotic results on the regret, we show that the minimax regret is achieved by mere random guessing under fairly mild conditions on the set of allowable configurations of the two arms. That is, we show that for {\sl every} allocation rule and for {\sl every} $n$, there is a configuration such that the regret at time $n$ is at least 1 -- $\epsilon$ times the regret of random guessing, where $\epsilon$ is any small positive constant.

A note on the separability principle in economies with single-peaked

We consider the problem of allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. A rule that has played a central role in the analysis of the problem is the so-called uniform rule. Chun (2001) proves that the uniform rule is the only rule satisfying Pareto optimality, no-envy, separability, and continuity (with respect to the social endowment). We obtain an alternative characterization by using a weak replication-invariance condition, called duplication-invariance, instead of continuity. Furthermore, we prove that Pareto optimality, equal division lower bound, and separability imply no-envy. Using this result, we strengthen one of Chun's (2001) characterizations of the uniform rule by showing that the uniform rule is the only rule satisfying Pareto optimality, equal división lower bound, separability, and either continuity or duplication-invariance.

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.

Statistical treatment of grain-size curves and empirical distributions: densities as compositions?

The preceding two editions of CoDaWork included talks on the possible considerationof densities as infinite compositions: Egozcue and D´ıaz-Barrero (2003) extended theEuclidean structure of the simplex to a Hilbert space structure of the set of densitieswithin a bounded interval, and van den Boogaart (2005) generalized this to the setof densities bounded by an arbitrary reference density. From the many variations ofthe Hilbert structures available, we work with three cases. For bounded variables, abasis derived from Legendre polynomials is used. For variables with a lower bound, westandardize them with respect to an exponential distribution and express their densitiesas coordinates in a basis derived from Laguerre polynomials. Finally, for unboundedvariables, a normal distribution is used as reference, and coordinates are obtained withrespect to a Hermite-polynomials-based basis.To get the coordinates, several approaches can be considered. A numerical accuracyproblem occurs if one estimates the coordinates directly by using discretized scalarproducts. Thus we propose to use a weighted linear regression approach, where all k-order polynomials are used as predictand variables and weights are proportional to thereference density. Finally, for the case of 2-order Hermite polinomials (normal reference)and 1-order Laguerre polinomials (exponential), one can also derive the coordinatesfrom their relationships to the classical mean and variance.Apart of these theoretical issues, this contribution focuses on the application of thistheory to two main problems in sedimentary geology: the comparison of several grainsize distributions, and the comparison among different rocks of the empirical distribution of a property measured on a batch of individual grains from the same rock orsediment, like their composition

A note on discrete claims problems

In this note, we consider claims problems with indivisible goods. Specifically, by applying recursively the P-rights lower bound (Jiménez-Gómez and Marco-Gil (2008)), we ensure the fulfillment of Weak Order Preservation, considered by many authors as a minimal requirement of fairness. Moreover, we retrieve the Discrete Constrained Equal Losses and the Discrete Constrained Equal Awards rules (Herrero and Martíınez (2008)). Finally, by the recursive double imposition of a lower and an upper bound, we obtain the average between them. Keywords: Claims problems, Indivisibilities, Order Preservation, Constrained Egalitarian rules, Midpoint. JEL classification: C71, D63, D71.

Fair bounds based solidarity

How should scholarships be distributed among the (public) higher education students? We raise this situation as a redistribution problem. Following the approach developed in Fleurbaey (1994) and Bossert (1995), redistribution should be based on the notion of solidarity and it re-allocates resources taking into account only agents’ relevant characteristics. We also follow Luttens (2010a), who considers that compensation of relevant characteristics must be based on a lower bound on what every individual deserves. In doing so, we use the so-called fair bound (Moulin (2002)) to define an egalitarian redistribution mechanism and characterize it in terms of non-negativity, priority in lower bound and solidarity. Finally, we apply our approach to the scholarships redistribution problem. Keywords: Redistribution mechanism, Lower bounds, Scholarship, Solidarity. JEL classification: C71, D63, D71

A note on Luttens’ Minimal rights based solidarity

Following the approach developed by Luttens (2010), we consider a model where individuals with di fferent levels of skills exert di fferent levels of e ffor. Speci fically, we propose a redistribution mechanism based on a lower bound on what every individual deserves: the so-called minimal rights (O'Neill (1982)). Our re finement of Luttens' mechanism ensures at the same time minimal rights based solidarity, participation (non-negativity) and claims feasibility. Keywords: Redistribution mechanism, Minimal rights, Solidarity, Participation, Claims feasibility. JEL classi fication: C71, D63, D71.

Interplay of spectral efficiency, power and doppler spectrum for reference-signal-assisted wireless communication

Expressions relating spectral efficiency, power, and Doppler spectrum, are derived for Rayleigh-faded wireless channels with Gaussian signal transmission. No side information on the state of the channel is assumed at the receiver. Rather, periodic reference signals are postulated in accordance with the functioning of most wireless systems. The analysis relies on a well-established lower bound, generally tight and asymptotically exact at low SNR. In contrast with most previous studies, which relied on block-fading channel models, a continuous-fading model is adopted. This embeds the Doppler spectrum directly in the derived expressions, imbuing them with practical significance. Closed-form relationships are obtained for the popular Clarke-Jakes spectrum and informative expansions, valid for arbitrary spectra, are found for the low- and high-power regimes. While the paper focuses on scalar channels, the extension to multiantenna settings is also discussed.

What is the value of joint processing of pilots and data in block-fading channels?

The spectral efficiency achievable with joint processing of pilot and data symbol observations is compared with that achievable through the conventional (separate) approach of first estimating the channel on the basis of the pilot symbols alone, and subsequently detecting the datasymbols. Studied on the basis of a mutual information lower bound, joint processing is found to provide a non-negligible advantage relative to separate processing, particularly for fast fading. It is shown that, regardless of the fading rate, only a very small number of pilot symbols (at most one per transmit antenna and per channel coherence interval) shouldbe transmitted if joint processing is allowed.

Compensating wage differentials and voluntary job changes: Evidence from West Germany

Does the labor market place wage premia on jobs that involve physical strain,job, insecurity or bad regulation of hours? This paper derives bounds on themonetary returns to these job disamenities in the West German labor market.We show that in a market with dispersion in both job characteristics andwages, the average wage change of workers who switch jobs voluntarily and optfor consuming more (less) disamenities,provides an upper (lower) bound on themarket return to the disamenity. Using longitudinal information from workersin the German Socio Economic Panel, we estimate an upper bound of 5% and alower bound of 3.5% for the market return to work strain in a job.

Existence of sparsely supported correlated equilibria

We show that every finite N-player normal form game possesses a correlated equilibrium with a precise lower bound on the number of outcomes to which it assigns zero probability. In particular, the largest games with a unique fully supported correlated equilibrium are two-player games; moreover, the lower bound grows exponentially in the number of players N.