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This study gives detailed proofs of some limit theorems in probability which are important in theoretical and applied probability, The general introduction contains definitions and theorems that are basic tools of the later development. Included in this first chapter is material concerning normal distributions and characteristic functions, The second chapter introduces lower and upper bounds of the ratio of the binomial distribution to the normal distribution., Then these bound are used to prove the local Deioivre-Laplace limit theorem. The third chapter includes proofs of the central limit theorems for identically distributed and non-identically distributed random variables,

Since Benoit Mandelbrot's discovery of the fractal nature of financial price series, the quantitative analysis of financial markets has been an area of increasing interest for scientists, traders, and regulators. Further, major technological advances over this time have facilitated not only financial innovations, but also the computational ability to analyze and model markets. The stylized facts are qualitative statistical signatures of financial market data that hold true across different stocks and over many different timescales. In pursuit of a mechanistic understanding of markets, we look to accurately quantify such statistics. With this quantification, we can test computational market models against the stylized facts and run controlled experiments. This requires both discovery of new stylized facts, and a persistent testing of old stylized facts on new data. Using NASDAQ provided data covering the years 2008-2009, we analyze the trades of 120 stocks. An analysis of the stylized facts guides our exploration of the data, where our results corroborate other findings in the existing body of literature. In addition, we search for statistical indicators of market instability in our data sets. We find promising areas for further study, and obtain three key results. Throughout our sample data, high frequency trading plays a larger role in rapid price changes of all sizes than would be randomly expected, but plays a smaller role than usual during rapid price changes of large magnitude. Our analysis also yields further evidence of the long term persistence in the autocorrelations of signed order flow, as well as evidence of long range dependence in price returns.

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Three probabilistic methods of different complexity for slope stability calculations are in this work evaluated with respect to a well-documented case study of slope failure in Lodalen, Norway. A finite element method considering spatial random fields of uncorrelated parameters cohesion and friction angle is taken as a reference for comparison with two simpler methods based on Taylor series expansion, known as first-order-second-moment methods. It is shown that the FOSM method enhanced by a reduction of variance of input parameters due to spatial averaging along the potential failure surface leads to a significant improvement in predictions. This method is computationally inexpensive and can be used in combination with any existing finite element code, it is thus a useful approximate probabilistic method for geotechnical practice. Several limitations of the extended method for calculating probability of a slope failure are identified. An advanced hypoplastic constitutive model is used in probabilistic analyses of a typical geotech- nical problem, strip footing. The model, including horizontal and vertical correlation lengths, was calibrated using a set of experimental data on sand from horizontally stratified deposit. Some parameters followed normal, whereas other followed lognormal distributions....

Digitized by Kansas Correctional Industries

Digitized by Kansas Correctional Industries