Limit theorems and approximations with applications to insurance risk and queueing theory

The analysis of ruin probabilities constitutes a cornerstone of insurance risk theory. This dissertation develops limit theorems and approximations that can be used to obtain the probability that an insurer faces eventual ruin in both the case of zero interest rates and when of stochastic return on investments are present. This theory is also relevant in the analysis of the single most important model in queueing theory, namely the single-server queue. Other applications to statistical sequential analysis and time series analysis are also discussed.; To deal with the case of zero interest rates, we derive corrected diffusion approximations for ladder height related quantities of a general random walk with small negative drift. This theory generalizes and extends important results in the literature, including results due to Siegmund (1979), Chang (1992), and Chang and Peres (1997). As it is discussed in the cited references, these types of corrected diffusion approximations have potential applications not only to insurance risk and queueing theory, but also to statistical sequential analysis among other applied disciplines.; In addition, we provide new tools for the probabilistic analysis of systems with heavy tailed characteristics. In particular, accurate approximations for the ruin probability, (in diffusion scale) under heavy tailed claims, follow from new asymptotic expansions for geometric sums that are also introduced in this dissertation. These asymptotic expansions are also applied to the analysis of defective renewal equations, which in turn yields new corrected diffusion approximations for M/G/c queues and so-called perturbed risk models (of the type of Dufresne and Gerber (1991)).; When stochastic interest rates are considered, it turns out that one must study the distribution of so-called perpetuities or infinite horizon discounted rewards. Such perpetuities are also of interest in mathematical finance and time series analysis (in the context of ARCH models). We develop approximations and computational algorithms to estimate the distribution of such infinite horizon discounted rewards. These approximations include laws of large numbers, central limit theorems, Edgeworth expansions, large deviation principles, and sharp tail asymptotics.