The Busy Beaver Competition: a historical survey

Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3, ... More and more powerful computers have made possible the computation of lower bounds for these values. In 1988, Brady extended the definitions to functions on two variables. We give a historical survey of these works. The successive record holders in the Busy Beaver Competition are displayed, with their discoverers, the date they were found, and, for some of them, an analysis of their behavior.Comment: 70 page

Problems in number theory from busy beaver competition

By introducing the busy beaver competition of Turing machines, in 1962, Rado defined noncomputable functions on positive integers. The study of these functions and variants leads to many mathematical challenges. This article takes up the following one: How can a small Turing machine manage to produce very big numbers? It provides the following answer: mostly by simulating Collatz-like functions, that are generalizations of the famous 3x+1 function. These functions, like the 3x+1 function, lead to new unsolved problems in number theory.Comment: 35 page

Rates of convergence for robust geometric inference

Distances to compact sets are widely used in the field of Topological Data Analysis for inferring geometric and topological features from point clouds. In this context, the distance to a probability measure (DTM) has been introduced by Chazal et al. (2011) as a robust alternative to the distance a compact set. In practice, the DTM can be estimated by its empirical counterpart, that is the distance to the empirical measure (DTEM). In this paper we give a tight control of the deviation of the DTEM. Our analysis relies on a local analysis of empirical processes. In particular, we show that the rates of convergence of the DTEM directly depends on the regularity at zero of a particular quantile fonction which contains some local information about the geometry of the support. This quantile function is the relevant quantity to describe precisely how difficult is a geometric inference problem. Several numerical experiments illustrate the convergence of the DTEM and also confirm that our bounds are tight

An Empirical Analysis of U.S. Aggregate Portfolio Allocations

This paper analyzes the important time variation in U.S. aggregate portfolio allocations. To do so, we first use flexible descriptions of preferences and investment opportunities to derive optimal decision rules that nest tactical, myopic, and strategic portfolio allocations. We then compare these rules to the data through formal statistical analysis. Our main results reveal that i) purely tactical and myopic investment behaviors are unambiguously rejected, ii) strategic portfolio allocations are strongly supported, and iii) the Fama-French factors best explain empirical portfolio shares.portfolio; factorial pricing; dynamic hedging

An Empirical Analysis of U.S. Aggregate Portfolio Allocations

This paper analyzes the important time variation in U.S. aggregate portfolio allocations. To do so, we first use flexible descriptions of preferences and investment opportunities to derive optimal decision rules that nest tactical, myopic, and strategic portofolio allocations. We then compare these rules to the data through formal statistical analysis. Our main results reveal that i) purely tactical and myopic investment behaviors are unambiguously rejected, ii) strategic portfolio allocations are strongly supported, and iii) the Fama-French factors best explain empirical portfolios shares.Dynamic Hedging, Risk Aversion, Inter-temporal Substitution, Time-Varying Investment Opportunity Set

Recursive Measures of Total Wealth and Portfolio Return

This letter presents and assesses a procedure to generate recursive measures of aggregate total wealth and portfolio return. Conceptually, the procedure is more flexible than the classical replacement cost and present value methods. Empirically, the procedure yields recursive measures that appear more realistic than those obtained from the classical methods.Financial, human, and tangible assets; present value and replacement cost methods

The Taylor principle and global determinacy in a non Ricardian world

The Taylor principle is quite usually considered as a central condition for price determinacy. Recently, however, this has been questioned on several grounds, notably because (i) this condition is a condition for local determinacy, not global determinacy (ii) it has been derived in "Ricardian" economies, and it appears that going to a non-Ricardian framework makes a very big difference for the determinacy conditions. In this paper we scrutinize the two issues together, and we find that for non-Ricardian equilibria the Taylor principle is replaced by another "financial dominance" criterion.Taylor principle ; Taylor rules ; global determinacy ; price determinacy ; non Ricardian economies ; non Ricardian equilibria

An Empirical Analysis of U.S. Aggregate Portfolio Allocations

This paper analyzes the important time variation in U.S. aggregate portfolio allocations. To do so, we first use flexible descriptions of preferences and investment opportunities to derive optimal decision rules that nest tactical, myopic, and strategic portfolio allocations. We then compare these rules to the data through formal statistical analysis. Our main results reveal that i) purely tactical and myopic investment behaviors are unambiguously rejected, ii) strategic portfolio allocations are strongly supported, and iii) the Fama-French factors best explain empirical portfolio shares.Dynamic Hedging; Risk Aversion; Inter-temporal Substitution; Time-Varying Investment Opportunity Set.

An Empirical Analysis of U.S. Aggregate Portfolio Allocations

This paper analyzes the important time variation in U.S. aggregate portfolio allocations. To do so, we first use flexible descriptions of preferences and investment opportunities to derive optimal decision rules that nest tactical, myopic, and strategic portfolio allocations. We then compare these rules to the data through formal statistical analysis. Our main results reveal that i) purely tactical and myopic investment behaviors are unambiguously rejected, ii) strategic portfolio allocations are strongly supported, and iii) the Fama-French factors best explain empirical portfolio shares. Ce papier analyse la forte variation chronologique dans les portefeuilles agrégés américains. À cet effet, nous utilisons des descriptions flexibles des préférences et des opportunités d'investissement afin de dériver les allocations tactiques, myopes et stratégiques. Ces règles sont ensuite comparées aux données dans le cadre d'une analyse statistique formelle. Nos principaux résultats révèlent que i) les règles purement myopes ou tactiques sont rejetées, ii) les portefeuilles stratégiques sont supportés et iii) les facteurs Fama-French sont ceux qui reproduisent le mieux les allocations empiriques.factorial models of returns, myopic and strategic, non-expected utility, tactical portfolio allocations , modèles factoriels des rendements, myopes et stratégiques, portefeuilles tactiques, utilité non espérée