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The argument that I tried to elaborate on in this paper is that the conceptual problem behind the traditional competence/performance distinction does not go away, even if we abandon its original Chomskyan formulation. It returns as the question about the relation between the model of the grammar and the results of empirical investigations – the question of empirical verification The theoretical concept of markedness is argued to be an ideal correlate of gradience. Optimality Theory, being based on markedness, is a promising framework for the task of bridging the gap between model and empirical world. However, this task not only requires a model of grammar, but also a theory of the methods that are chosen in empirical investigations and how their results are interpreted, and a theory of how to derive predictions for these particular empirical investigations from the model. Stochastic Optimality Theory is one possible formulation of a proposal that derives empirical predictions from an OT model. However, I hope to have shown that it is not enough to take frequency distributions and relative acceptabilities at face value, and simply construe some Stochastic OT model that fits the facts. These facts first of all need to be interpreted, and those factors that the grammar has to account for must be sorted out from those about which grammar should have nothing to say. This task, to my mind, is more complicated than the picture that a simplistic application of (not only) Stochastic OT might draw.
Approximating Perpetuities
(2006)
A perpetuity is a real valued random variable which is characterised by a distributional fixed-point equation of the form X=AX+b, where (A,b) is a vector of random variables independent of X, whereas dependencies between A and b are allowed. Conditions for existence and uniqueness of solutions of such fixed-point equations are known, as is the tail behaviour for most cases. In this work, we look at the central area and develop an algorithm to approximate the distribution function and possibly density of a large class of such perpetuities. For one specific example from the probabilistic analysis of algorithms, the algorithm is implemented and explicit error bounds for this approximation are given. At last, we look at some examples, where the densities or at least some properties are known to compare the theoretical error bounds to the actual error of the approximation. The algorithm used here is based on a method which was developed for another class of fixed-point equations. While adapting to this case, a considerable improvement was found, which can be translated to the original method.