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<p>This dissertation includes my research works during Ph.D. career about three different kinds of random walks in (dynamical) random environments. It includes my two published papers “Functional weak limit of random walks in cooling random environments” which has been published in electronic communications in probability in 2020, and “Variable speed symmetric random walk driven by the simple symmetric exclusion process” which is the joint work with Peterson and Menezes and has been published in electronic journals of probability in 2021. This dissertation also includes my two other projects, one is the joint work with Janjigian and Emrah about moderate deviation and exit time estimates in integrable directed polymer models. The other one is the joint work with Peterson and Conrado that extends the weak limit of random walks in cooling randon environments with underlying environment is in the transient case.</p>

Master of Science / Department of Statistics / Paul I. Nelson / The variance of a response in a one-way random effects model can be expressed as the sum of the variability among and within treatment levels. Conventional methods of statistical analysis for these models are based on the assumption of normality of both sources of variation. Since this assumption is not always satisfied and can be difficult to check, it is important to explore the performance of normal based inference when normality does not hold. This report uses simulation to explore and assess the robustness of the F-test for the presence of an among treatment variance component and the normal theory confidence interval for the intra-class correlation coefficient under several non-normal distributions. It was found that the power function of the F-test is robust for moderately heavy-tailed random error distributions. But, for very heavy tailed random error distributions, power is relatively low, even for a large number of treatments. Coverage rates of the confidence interval for the intra-class correlation coefficient are far from nominal for very heavy tailed, non-normal random effect distributions.

published_or_final_version / Computer Science / Master / Master of Philosophy

Random number generators (RNGs) are widely used in conducting Monte Carlo simulation studies, which are important in the field of statistics for comparing power, mean differences, or distribution shapes between statistical approaches. Statistical results, however, may differ when different random number generators are used. Often older methods have been blindly used with no understanding of their limitations. Many random functions supplied with computers today have been found to be comparatively unsatisfactory. In this study, five multiplicative linear congruential generators (MLCGs) were chosen which are provided in the following statistical packages: RANDU (IBM), RNUN (IMSL), RANUNI (SAS), UNIFORM(SPSS), and RANDOM (BMDP). Using a personal computer (PC), an empirical investigation was performed using five criteria: period length before repeating random numbers, distribution shape, correlation between adjacent numbers, density of distributions and normal approach of random number generator (RNG) in a normal function. All RNG FORTRAN programs were rewritten into Pascal which is more efficient language for the PC. Sets of random numbers were generated using different starting values. A good RNG should have the following properties: a long enough period; a well-structured pattern in distribution; independence between random number sequences; random and uniform distribution; and a good normal approach in the normal distribution. Findings in this study suggested that the above five criteria need to be examined when conducting a simulation study with large enough sample sizes and various starting values because the RNG selected can affect the statistical results. Furthermore, a study for purposes of indicating reproducibility and validity should indicate the source of the RNG, the type of RNG used, evaluation results of the RNG, and any pertinent information related to the computer used in the study. Recommendations for future research are suggested in the area of other RNGs and methods not used in this study, such as additive, combined, mixed and shifted RNGs.

We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk. We call this the range of the random walk. We derive a large deviation principle for the probability that the range of simple random walk deviates from its mean. Our result describes the behaviour for deviation below the typical value. This is a result analogous to that obtained by van den Berg, Bolthausen, and den Hollander for the volume of the Wiener sausage. In the second part of the thesis, we are interested in the number of lattice sites visited by two independent simple random walks starting at the origin. We call this the intersection of ranges. We derive a large deviation principle for the probability that the intersection of ranges by time n exceeds a multiple of n. This is also an analogous result of the intersection volume of two independent Wiener sausages.

In the first part of this thesis I consider site and bond percolation on a Random Connection Model and prove that for a wide range of connection functions the critical site probability is strictly greater than the critical bond probability and use this fact to improve previously known non-strict inequalities to strict inequalities. In the second part I consider percolation on the even phase of a Random Sequential Adsorption model and prove that the critical intensity is finite and strictly bigger than 1. Both of these main results make use of an enhancement technique.

Adaptive random testing (ART) has been proposed as an enhancement to random testing for situations where failure-causing inputs are clustered together. The basic idea of ART is to evenly spread test cases throughout the input domain. It has been shown by simulations and empirical analysis that ART frequently outperforms random testing. However, there are some outstanding issues on the cost-effectiveness and practicality of ART, which are the main foci of this thesis. Firstly, this thesis examines the basic factors that have an impact on the faultdetection effectiveness of adaptive random testing, and identifies favourable and unfavourable conditions for ART. Our study concludes that favourable conditions for ART occur more frequently than unfavourable conditions. Secondly, since all previous studies allow duplicate test cases, there has been a concern whether adaptive random testing performs better than random testing because ART uses fewer duplicate test cases. This thesis confirms that it is the even spread rather than less duplication of test cases which makes ART perform better than RT. Given that the even spread is the main pillar of the success of ART, an investigation has been conducted to study the relevance and appropriateness of several existing metrics of even spreading. Thirdly, the practicality of ART has been challenged for nonnumeric or high dimensional input domains. This thesis provides solutions that address these concerns. Finally, a new problem solving technique, namely, mirroring, has been developed. The integration of mirroring with adaptive random testing has been empirically shown to significantly increase the cost-effectiveness of ART. In summary, this thesis significantly contributes to both the foundation and the practical applications of adaptive random testing.

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