Discontinuities in the Distribution of Great Wealth: Sectoral Forces Old and New

National surveys of household economics and well-being in the United States usually focus on income. In those income surveys with supplemental wealth modules, the very rich are underrepresented if not unrepresented. Typically, wealth data are truncated such that they do not afford a view of the extreme top of the distribution. Therefore, we attempt to supplement our knowledge about elite wealth holdings by compiling data on the richest individuals and families in the United States. To do so, we draw from the rosters of the "Forbes Four Hundred," which have been published annually by Forbes magazine since 1982. Along with information from other business press reports and standard biographical sources, rosters of the very rich enable research on inequality at the extreme of the wealth distribution during a period of dramatic change in the composition and concentration of wealth. In this study, we focus analytically on economic sectors because we are interested less in the maldistribution of wealth by demographic groups than in inequality between different economic sectors. We will first specify our analytical approach, then examine issues in the use of business press rosters of the very rich as a data source, and follow with a discussion of the dimensions and categories of our sector typology. After presenting our results, we will address how sectoral forces old and new affect economic opportunity and great wealth outcomes.

Signal propagation and noisy circuits

The information carried by a signal decays when the signal is corrupted by random noise. This occurs when a message is transmitted over a noisy channel, as well as when a noisy component performs computation. We first study this signal decay in the context of communication and obtain a tight bound on the rate at which information decreases as a signal crosses a noisy channel. We then use this information theoretic result to obtain depth lower bounds in the noisy circuit model of computation defined by von Neumann. In this model, each component fails (produces 1 instead of 0 or vice-versa) independently with a fixed probability, and yet the output of the circuit is required to be correct with high probability. Von Neumann showed how to construct circuits in this model that reliably compute a function and are no more than a constant factor deeper than noiseless circuits for the function. We provide a lower bound on the multiplicative increase in circuit depth necessary for reliable computation, and an upper bound on the maximum level of noise at which reliable computation is possible

On the maximum tolerable noise of k-input gates for reliable computation by formulas

We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions

Information Theory and Noisy Computation

We report on two types of results. The first is a study of the rate of decay of information carried by a signal which is being propagated over a noisy channel. The second is a series of lower bounds on the depth, size, and component reliability of noisy logic circuits which are required to compute some function reliably. The arguments used for the circuit results are information-theoretic, and in particular, the signal decay result is essential to the depth lower bound. Our first result can be viewed as a quantified version of the data processing lemma, for the case of Boolean random variables