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This paper aims at analyzing a macro economy with a continuum of infinitely-lived households that make rational decisions about consumption and wealth savings in the face of employment and aggregate productivity shocks. The heterogeneous population structure arises when households differ in wealth and employment status against which they cannot insure. In this framework, the household wealth evolution is modeled as a mixture Markov process. The stationary wealth distributions are obtained using eigen structures of transition matrices under the Perron-Frobenius theorem. This step is utilized repeatedly to find the equilibrium state of the system, and it leads to an efficient framework for studying the dynamic general equilibrium. A systematic evaluation of the equilibrium state under different initial conditions is further presented and analyzed. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Summer Semester, 2010. / Date of Defense: April 28, 2010. / Markov Process, The Perron-Frobenius Theorem, Wealth Distributions, Dynamic General Equilibrium Modeling / Includes bibliographical references. / Anuj Srivastava, Professor Co-Directing Dissertation; Paul Beaumont, Professor Co-Directing Dissertation; Wei Wu, Committee Member; Alec Kercheval, Outside Committee Member.

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It is now common to hold that causes do not always (and perhaps never) determine their effects, and indeed theories of "probabilistic causation" abound. The basic idea of these theories is that C causes E just in case C and E both occur, and the chance of E would have been lower than it is had C not occurred. The problems with these accounts are that (i) the notion of chance remains primitive, and (ii) this account of causation does not coincide with the intuitive notion of causation as ontological support. Turning things around, I offer an analysis of chance in terms of causation, called the causal theory of chance. The chance of an event E is the degree to which it is determined by its causes. Thus chance events have full causal pedigrees, just like determined events; they are not "events from nowhere". I hold that, for stochastic as well as for deterministic processes, the actual history of a system is caused by its dynamical properties (represented by the lagrangian) and the boundary condition. A system is stochastic if (a description of) the actual history is not fully determined by maximal knowledge of these causes, i.e. it is not logically entailed by them. If chance involves partial determination, and determination is logical entailment, then there must be such a thing as partial entailment, or logical probability. To make the notion of logical probability plausible, in the face of current opposition to it, I offer a new account of logical probability which meets objections levelled at the previous accounts of Keynes and Carnap. The causal theory of chance, unlike its competitors, satisfies all of the following criteria: (i) Chance is defined for single events. (ii) Chance supervenes on the physical properties of the system in question. (iii) Chance is a probability function, i.e. a normalised measure. (iv) Knowledge of the chance of an event warrants a numerically equal degree of belief, i.e. Miller's Principle can be derived within the theory. (v) Chance is empirically accessible, within any given range of error, by measuring relative frequencies. (vi) With an additional assumption, the theory entails Reichenbach's Common Cause Principle (CCP). (vii) The theory enables us to make sense of probabilities in quantum mechanics. The assumption used to prove the CCP is that the state of a system represents complete information, so that the state of a composite system "factorises" into a logical conjunction of states for the sub-systems. To make sense of quantum mechanics, particularly the EPR experiment, we drop this assumption. In this case, the EPR criterion of reality is false. It states that if an event E is predictable, and locally caused, then it is locally predictable. This fails when maximal information about a pair of systems does not factorise, leading to a non-locality of knowledge.

In this thesis we are concerned with estimation of clustering probabilities for univariate heavy tailed time series. We employ functional convergence of a bivariate tail empirical process to conclude asymptotic normality of an estimator of the clustering probabilities. Theoretical results are illustrated by simulation studies.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1962. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 75-79).

Thesis (Ph. D.)--University of Illinois, 1929. / Vita. "Reprinted from the Annals of mathematical statistics, v.1, May, 1930,p. 137-158."

From Introduction: In probability theory it is often convenient to represent laws by characteristic functions, these being particularly suited to classical analysis. Trotter has suggest ted that probability laws can also be represented by probability operators. These operators are easily handled since they are continuous, and hence bounded, positive linear operators on a normed linear space. This representation arises because distribution functions and their complete convergence correspond to probability operators and their complete convergence. Hence the relations between distribution functions and probability operators will be discussed before the introduction of probability laws.

In this thesis we are concerned with estimation of clustering probabilities for univariate heavy tailed time series. We employ functional convergence of a bivariate tail empirical process to conclude asymptotic normality of an estimator of the clustering probabilities. Theoretical results are illustrated by simulation studies.