C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsOn spectral simulation of fractional Brownian motion . Despite the availability of several exact simulation methods, attention has been paid to approximate simulation (i.e., the output is approximately fBm), particularly because of possible time savings. In this paper, we study the class of approximate methods that are based on the spectral properties of fBm's stationary incremental process, usually called fractional Gaussian noise (fGn). The main contribution is a proof of asymptotical exactness (in a sense that is made precise) of these spectral methods. Moreover, we establish the connection between the spectral simulation approach and a widely used method, originally proposed by Paxson, that lacked a formal mathematical justification. The insights enable us to evaluate the Paxson method in more detail. It is also shown that spectral simulation is related to the fastest known exact method. Mathematics Subject Classification: 65C05, 62M15

For a given one-dimensional random walk {Sn} with a subexponential step-size distribution, we present a unifying theory to study the sequences {xn} for which P{Sn > x} ∼ nP{S1 > x} as n → ∞ uniformly for x ≥ xn. We also investigate the stronger "local" analogue,Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory.When specialized to the most important subclasses of subexponential distributions that have been studied in the literature, we reproduce known theorems and we supplement them with new results.

Consider a centered separable Gaussian process Y with a variance function that is regularly varying at infinity with index 2H 2 ð0; 2Þ: Let f be a 'drift' function that is strictly increasing, regularly varying at infinity with index b4H; and vanishing at the origin. Motivated by queueing and risk models, we investigate the asymptotics for u ! 1 of the probability Pðsup tX0 Y t À fðtÞ4uÞ as u ! 1: To obtain the asymptotics, we tailor the celebrated double sum method to our general framework. Two different families of correlation structures are studied, leading to four qualitatively different types of asymptotic behavior. A generalized Pickands' constant appears in one of these cases. Our results cover both processes with stationary increments (including Gaussian integrated processes) and self-similar processes.

Abstract. We propose an exact simulation method for Brown-Resnick random fields, building on new representations for these stationary max-stable fields. The main idea is to apply suitable changes of measure.