Slow Convergence in Generalized Central Limit Theorems

We study the central limit theorem in the non-normal domain of attraction to symmetric $\alpha$-stable laws for $0<\alpha\leq2$. We show that for i.i.d. random variables $X_i$, the convergence rate in $L^\infty$ of both the densities and distributions of $\sum_i^n X_i/(n^{1/\alpha}L(n))$ is at best logarithmic if $L$ is a non-trivial slowly varying function. Asymptotic laws for several physical processes have been derived using central limit theorems with $\sqrt{n\log n}$ scaling and Gaussian limiting distributions. Our result implies that such asymptotic laws are accurate only for exponentially large $n$.Comment: To appear in Comptes Rendus de l'Acad\'emie des Sciences, Math\'ematique

Approximate, not perfect synchrony maximizes the downstream effectiveness of excitatory neuronal ensembles

The most basic functional role commonly ascribed to synchrony in the brain is that of amplifying excitatory neuronal signals. The reasoning is straightforward: When positive charge is injected into a leaky target neuron over a time window of positive duration, some of it will have time to leak back out before an action potential is triggered in the target, and it will in that sense be wasted. If the goal is to elicit a firing response in the target using as little charge as possible, it seems best to deliver the charge all at once, i.e., in perfect synchrony. In this article, we show that this reasoning is correct only if one assumes that the input ceases when the target crosses the firing threshold, but before it actually fires. If the input ceases later-for instance, in response to a feedback signal triggered by the firing of the target-the "most economical" way of delivering input (the way that requires the least total amount of input) is no longer precisely synchronous, but merely approximately so. If the target is a heterogeneous network, as it always is in the brain, then ceasing the input "when the target crosses the firing threshold" is not an option, because there is no single moment when the firing threshold is crossed. In this sense, precise synchrony is never optimal in the brain.R01 NS067199 - NINDS NIH HH

When are signals complements or substitutes?

The paper introduces a notion of complementarity (substitutability) of two signals which requires that in all decision problems each signal becomes more (less) valuable when the other signal becomes available. We provide a general characterization which relates complementarity and substitutability to a Blackwell-comparison of two auxiliary signals. In a special setting with a binary state space and binary, symmetric signals, we find an explicit characterization that permits an intuitive interpretation of complementarity and substitutability. We demonstrate how these conditions extend to the general case. Finally, we study implications of complementarity and substitutability for information acquisition and in a second price auction

Infinitesimal phase response functions can be misleading

Phase response functions are the central tool in the mathematical analysis of pulse-coupled oscillators. When an oscillator receives a brief input pulse, the phase response function specifies how its phase is altered, as a function of the phase at which the input arrives. When the pulse is weak, it is customary to linearize the dependence of the response on pulse strength. The result is called the {\em infinitesimal} phase response function. These ideas have been used extensively in theoretical biology and also in some areas of engineering. I give an example in which the infinitesimal phase response function predicts that two oscillators, as they exchange pulses back and fourth, will converge to synchrony. Using the true phase response, I prove this prediction to be false for all positive interaction strengths. For short, the analogue of the Hartman-Grobman theorem that one might, at first sight, expect to hold is invalid

When are Signals Complements or Substitutes?

The paper introduces a notion of complementarity (substitutability) of two signals which requires that in all decision problems each signal becomes more (less) valuable when the other signal becomes available. We provide a general characterization which relates com- plementarity and substitutability to a Blackwell comparison of two auxiliary signals. In a setting with a binary state space and binary signals, we find an explicit characteriza- tion that permits an intuitive interpretation of complementarity and substitutability. We demonstrate how these conditions extend to more general settings.Information, signals, complementarity, substitutability.

Expedient and Monotone Learning Rules

This paper considers learning rules for environments in which little prior and feedback information is available to the decision-maker. Two properties of such learning rules, absolute expediency and monotonicity, are studied. The paper provides some necessary and some sufficient conditions for these properties. A number of examples show that there is quite a large variety of learning rules which have these properties. It is also shown that all learning rules that have these properties are, in some sense, related to replicator dynamics of evolutionary game theory.Absolute expediency, monotonicity, learning rule, decision making

An optimal Voting System when Voting is costly

We consider the design of an optimal voting system when voting is costly. For a private values model with two alternatives we show the optimality of a voting system that combines three elements: (i) there is an arbitrarily chosen default decision and non-participation is interpreted as a vote in favor of the default; (ii) voting is sequential; (iii) not all voters are invited to participate in the vote. We show the optimality of such a voting system by first arguing that it is first best, that is, it maximizes welfare when incentive compatibility constraints are ignored, and then showing that individual incentives and social welfare are sufficiently aligned to make the first best system incentive compatible. The analysis in this paper involves some methods that are new to the theory of mechanism design, and it is also a purpose of this paper to explore these new methods.Voting; mechanism design; committees.