Large deviations of stochastic systems and applications

<p>This dissertation focuses on large deviations of stochastic systems with applications to optimal control and system identification. It encompasses analysis of two-time-scale Markov processes and system identification with regular and quantized data. First, we develops large deviations principles for systems driven by continuous-time Markov chains with twotime scales and related optimal control problems. A distinct feature of our setup is that the Markov chain under consideration is time dependent or inhomogeneous. The use of two time-scale formulation stems from the effort of reducing computational complexity in a wide variety of applications in control, optimization, and systems theory. Starting with a rapidly fluctuating Markovian system, under irreducibility conditions, both large deviations upper and lower bounds are established first for a fixed terminal time and then for time-varying dynamic systems. Then the results are applied to certain dynamic systems and LQ control problems.</p> <p>Second, we study large deviations for identifications systems. Traditional system identification concentrates on convergence and convergence rates of estimates in mean squares, in distribution, or in a strong sense. For system diagnosis and complexity analysis, however, it is essential to understand the probabilities of identification errors over a finite data window. This paper investigates identification errors in a large deviations framework. By considering both space complexity in terms of quantization levels and time complexity with respect to data window sizes, this study provides a new perspective to understand the fundamental relationship between probabilistic errors and resources that represent data sizes in computer algorithms, sample sizes in statistical analysis, channel bandwidths in communications, etc.</p> <p>This relationship is derived by establishing the large deviations principle for quantized identification that links binary-valued data at one end and regular sensors at the other. Under some mild conditions, we obtain large deviations upper and lower bounds. Our results accommodate</p> <p>independent and identically distributed noise sequences, as well as more general classes of mixing-type noise sequences. Numerical examples are provided to illustrate the theoretical results.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Open Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2012

Author dc:creator

• He, Qi

Contributors dc:contributor

• George Yin

• averaging principle, complexity comparison, Identification system, Large deviations, non-homogeneity, Two-time-scale system
• Mathematics

Repository record dc:identifier

https://digitalcommons.wayne.edu/oa_dissertations/509

OAI identifier oai:identifier

oai:digitalcommons.wayne.edu:oa_dissertations-1508