Recently, stochastic geometry has been shown to be a very powerful tool to model, analyze, and design networks with random topologies such as wireless ad hoc and sensor networks as well as multi-tier cellular networks. In stochastic geometry analysis, point processes are used to model the positions and the channel access behaviors of the nodes. The thesis develops analytical frameworks to characterize the performance of large-scale wireless networks with random topologies. In particular, I use stochastic geometry tools to model, analyze, and design ad hoc networks, star-connected sensor networks, and infrastructure-based two-tier cellular networks. I have optimized the tradeoff between outage probability and spatial frequency reuse efficiency in carrier sensing-multiple-access based ad hoc networks. I have developed a novel spectrum efficient design paradigm for star-connected wireless sensor networks. For downlink transmission in cellular networks with cognitive femto access points (FAPs), I have quantified the performance gain imposed by cognition and developed a paradigm to optimize the spectrum sensing threshold for cognitive FAPs. Finally, I have developed a novel modeling paradigm for uplink transmission in cellular networks and obtained simple expressions for network performance metrics including the outage probability and average rate. Furthermore, I have revealed a transition point in the behavior of uplink transmission in cellular networks that depends on the relative values of the network parameters.

Stochastic generation of daily precipitation at multiple sites is frequently needed to evaluate the long-term effects of hydrologic and climate-change in design and operation of water resources systems. Capturing the spatial dependence of precipitation at multiple sites into a stochastic model presents a great challenge because of the non-normal bivariate distributions of precipitation-amounts. Without normalizing the precipitation amounts, many models have attempted to establish spatial dependence through alternative methods that tended to be cumbersome. In contrast, representing precipitation in Gaussian fields provides a generic structure that is well-amenable to statistical analyses facilitating easy implementation of models. The thrust of this thesis is to generate normalized precipitation data and transform them back into the original domain for applications and analyses. A multivariate censored distribution (MCD) and a multivariate autoregressive censored process (MACP) are developed to formulate two weather generation (WG) models. Parameters of censored distributions were estimated by using the maximum likelihood method. To reduce the magnanimity in the number of parameters and their temporal variation, elements of covariance matrices of models were represented by periodic functions. The performance of models was evaluated by comparing discrepancies in attributes. Three performance measures (i.e., the coefficient of determination, the coefficient efficiency and the root mean square error) suggested that simulated data to be indistinguishable from the historical precipitation sequences. The models were implemented with other techniques to address the three most common problems encountered in daily precipitation records. The first implementation is related to simulation of precipitation at un-gauged sites using the WG-MACP model with general regression neural networks or Kriging methods. The second implementation was related to infilling of missing observations a using the WG-MCD and WG-MACP models with Gibbs sampling. The third implementation was related to downscaling of monthly and daily output of the Canadian regional climate model (CRCM) using traditional and parametric Delta change methods.

The hemp plant is an attractive source of raw material for multiple products. Processing hemp requires the separation of fibre and core components of the plant. Peel tests were conducted for hemp stems to evaluate the strength required to peel fibre from the core. The average peeling force for the Alyssa variety was 0.39 N and that for the USO-14 variety was 0.87 N. The Ising model was implemented to produce a stochast ic model. The simulated peel test behaved similarly to the experimental peel test. A discrete element model (DEM) of a planetary ball mill was developed to predict the energy requirement of grinding hemp for fibre. Hemp grinding tests were performed on variety USO-31 using a planetary ball mill for model calibration purposes. Power draw measurements increased linearly increasing at greater grinding speeds. The DEM approximated power draw with relative error below 10% for grinding speeds below 400 rpm.

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In this thesis we consider estimation of the tail index for heavy tailed stochastic volatility models with long memory. We prove a central limit theorem for a Hill estimator. In particular, it is shown that neither the rate of convergence nor the asymptotic variance is affected by long memory. The theoretical findings are verified by simulation studies.

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The asymptotic properties of discrete time stochastic systems operating under feedback is addressed. It is assumed that a Markov chain $ Phi$ evolving on Euclidean space exists, and that the input and output processes appear as functions of $ Phi$. The main objectives of the thesis are (i) to extend various asymptotic properties of Markov chains to hold for arbitrary initial distributions; and (ii) to develop a robustness theory for Markovian systems. / A condition called local stochastic controllability, a generalization of the concept of controllability from linear system theory, is introduced and is shown to be sufficient to ensure that the first objective is met. The second objective is explored by introducing a notion of convergence for stochastic systems and investigating the behavior of the invariant probabilities corresponding to a convergent sequence of stochastic systems. / These general results are applied to two previously unsolved problems: The asymptotic behavior of linear state space systems operating under nonlinear feedback, and the stability and asymptotic behavior of a class of random parameter AR (p) stochastic systems under optimal control.

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Graduation date: 1993