"Non-Self-Averaging in Macroeconomic Models: A Criticism of Modern Micro-founded Macroeconomics"

Using a simple stochastic growth model, this paper emonstrates that the coefficient of variation of aggregate output or GDP does not necessarily go to zero even if the number of sectors or economic agents goes to infinity. This phenomenon known as non-self-averaging implies that even if the number of economic agents is large, dispersion can remain significant, and, therefore, that we can not legitimately focus on the means of aggregate variables. It, in turn, means that the standard microeconomic foundations based on the representative agent has little value for they are expected to provide us with dynamics of the means of aggregate variables. The paper also shows that non-self-averaging emerges in some representative urn models. It suggests that non-self-averaging is not pathological but quite generic. Thus, contrary to the main stream view, micro-founded macroeconomics such as a dynamic general equilibrium model does not provide solid micro foundations.

The Lent Particle Method, Application to Multiple Poisson Integrals

We give a extensive account of a recent new way of applying the Dirichlet form theory to random Poisson measures. The main application is to obtain existence of density for thelaws of random functionals of L\'evy processes or solutions of stochastic differential equations with jumps. As in the Wiener case the Dirichlet form approach weakens significantly theregularity assumptions. The main novelty is an explicit formula for the gradient or for the "carr\'e du champ' on the Poisson space called the lent particle formula because based on adding a new particle to the system, computing the derivative of the functional with respect to this new argument and taking back this particle before applying the Poisson measure. The article is expository in its first part and based on Bouleau-Denis [12] with several new examples, applications to multiple Poisson integrals are gathered in the last part which concerns the relation with the Fock space and some aspects of the second quantization

Non-Self-Averaging in Macroeconomic Models: A Criticism of Modern Micro-founded Macroeconomics

Using a simple stochastic growth model, this paper demonstrates that the coefficient of variation of aggregate output or GDP does not necessarily go to zero even if the number of sectors or economic agents goes to infinity. This phenomenon known as non-self-averaging implies that even if the number of economic agents is large, dispersion can remain significant, and, therefore, that we can not legitimately focus on the means of aggregate variables. It, in turn, means that the standard microeconomic foundations based on the representative agent has little value for they are expected to provide us with dynamics of the means of aggregate variables. The paper also shows that non-self-averaging emerges in some representative urn models. It suggests that non-self-averaging is not pathological but quite generic. Thus, contrary to the main stream view, micro-founded macroeconomics such as a dynamic general equilibrium model does not provide solid micro foundations.

Producing power-law distributions and damping word frequencies with two-stage language models

Standard statistical models of language fail to capture one of the most striking properties of natural languages: the power-law distribution in the frequencies of word tokens. We present a framework for developing statisticalmodels that can generically produce power laws, breaking generativemodels into two stages. The first stage, the generator, can be any standard probabilistic model, while the second stage, the adaptor, transforms the word frequencies of this model to provide a closer match to natural language. We show that two commonly used Bayesian models, the Dirichlet-multinomial model and the Dirichlet process, can be viewed as special cases of our framework. We discuss two stochastic processes-the Chinese restaurant process and its two-parameter generalization based on the Pitman-Yor process-that can be used as adaptors in our framework to produce power-law distributions over word frequencies. We show that these adaptors justify common estimation procedures based on logarithmic or inverse-power transformations of empirical frequencies. In addition, taking the Pitman-Yor Chinese restaurant process as an adaptor justifies the appearance of type frequencies in formal analyses of natural language and improves the performance of a model for unsupervised learning of morphology.48 page(s

Eigenvalue fluctuations for random regular graphs

One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random regular graphs. Specifically, we prove limit theorems for the fluctuations of linear spectral statistics of random regular graphs. We find both universal and non-universal behavior. Our most important tool is Stein's method for Poisson approximation, which we develop for use on random regular graphs. This is my Ph.D. thesis, based on joint work with Ioana Dumitriu, Elliot Paquette, and Soumik Pal. For the most part, it's a mashed up version of arXiv:1109.4094, arXiv:1112.0704, and arXiv:1203.1113, but some things in here are improved or new. In particular, Chapter 4 goes into more detail on some of the proofs than arXiv:1203.1113 and includes a new section. See Section 1.3 for more discussion on what's new and who contributed to what.Comment: 103 pages; Ph.D. thesis at the University of Washington, 201