Large deviations for random walks under subexponentiality: the big-jump domain

For a given one-dimensional random walk $\{S_n\}$ with a subexponential step-size distribution, we present a unifying theory to study the sequences $\{x_n\}$ for which $\mathsf{P}\{S_n>x\}\sim n\mathsf{P}\{S_1>x\}$ as $n\to\infty$ uniformly for $x\ge x_n$. We also investigate the stronger "local" analogue, $\mathsf{P}\{S_n\in(x,x+T]\}\sim n\mathsf{P}\{S_1\in(x,x+T]\}$. 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.Comment: Published in at http://dx.doi.org/10.1214/07-AOP382 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Per-site occupancy in the discrete parking problem

We consider the classical discrete parking problem, in which cars arrive uniformly at random on any two adjacent sites out of n sites on a line. An arriving car parks if there is no overlap with previously parked cars, and leaves otherwise. This process continues until there is no more space available for cars to park, at which point we may compute the jamming density En/n, which represents the expected fraction of occupied sites. We extend the classical results by not just considering the total expected number of cars parked, but also the probability of each site being occupied by a car. This we then use to provide an alternative derivation of the jamming density

Instability of MaxWeight scheduling algorithms

MaxWeight scheduling algorithms provide an effective mechanism for achieving queue stability and guaranteeing maximum throughput in a wide variety of scenarios. The maximum-stability guarantees however rely on the fundamental premise that the system consists of a fixed set of sessions with stationary ergodic traffic processes. In the present paper we examine a scenario where the population of active sessions varies over time, as sessions eventually end while new sessions occasionally start. We identify a simple necessary and sufficient condition for stability, and show that MaxWeight policies may fail to provide maximum stability. The intuitive explanation is that these policies tend to give preferential treatment to flows with large backlogs, so that the rate variations of flows with smaller backlogs are not fully exploited. In the usual framework with a fixed collection of flows, the latter phenomenon cannot persist since the flows with smaller backlogs will build larger queues and gradually start receiving more service. With a dynamic population of flows, however, MaxWeight policies may constantly get diverted to arriving flows, while neglecting the rate variations of a persistently growing number of flows in progress with relatively small remaining backlogs. We also perform extensive simulation experiments to corroborate the analytical findings

Structural bias in population-based algorithms

Challenging optimisation problems are abundant in all areas of science and industry. Since the 1950s, scientists have responded to this by developing ever-diversifying families of 'black box' optimisation algorithms. The latter are designed to be able to address any optimisation problem, requiring only that the quality of any candidate solution can be calculated via a 'fitness function' specific to the problem. For such algorithms to be successful, at least three properties are required: (i) an effective informed sampling strategy, that guides the generation of new candidates on the basis of the fitnesses and locations of previously visited candidates; (ii) mechanisms to ensure efficiency, so that (for example) the same candidates are not repeatedly visited; and (iii) the absence of structural bias, which, if present, would predispose the algorithm towards limiting its search to specific regions of the solution space. The first two of these properties have been extensively investigated, however the third is little understood and rarely explored. In this article we provide theoretical and empirical analyses that contribute to the understanding of structural bias. In particular, we state and prove a theorem concerning the dynamics of population variance in the case of real-valued search spaces and a 'flat' fitness landscape. This reveals how structural bias can arise and manifest as non-uniform clustering of the population over time. Critically, theory predicts that structural bias is exacerbated with (independently) increasing population size, and increasing problem difficulty. These predictions, supported by our empirical analyses, reveal two previously unrecognised aspects of structural bias that would seem vital for algorithm designers and practitioners. Respectively, (i) increasing the population size, though ostensibly promoting diversity, will magnify any inherent structural bias, and (ii) the effects of structural bias are more apparent when faced with (many classes of) 'difficult' problems. Our theoretical result also contributes to the 'exploitation/exploration' conundrum in optimisation algorithm design, by suggesting that two commonly used approaches to enhancing exploration - increasing the population size, and increasing the disruptiveness of search operators - have quite distinct implications in terms of structural bias

Estimates for interval probabilities of the sums of random variables with locally subexponential distributions

Let {i} i=1 be a sequence of independent identically distributed nonnegative random variables, S n = ¿1 + ¿ +¿n. Let ¿ = (0, T] and x + ¿ = (x, x + T]. We study the ratios of the probabilities P(S n e x + ¿)/P(¿ 1 e x + ¿) for all n and x. The estimates uniform in x for these ratios are known for the so-called ¿-subexponential distributions. Here we improve these estimates for two subclasses of ¿-subexponential distributions; one of them is a generalization of the well-known class LC to the case of the interval (0, T] with an arbitrary T = 8. Also, a characterization of the class LC is given

Estimates for the distributions of the sums of subexponential random variables

Let be a random walk with independent identically distributed increments . We study the ratios of the probabilities P(S n >x) / P(1 > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(S > x) E P(1 > x) as x . Here is a positive integer-valued random variable independent of . The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process

Asymptotics for first-passage times of Lévy processes and random walks

We study the exact asymptotics for the distribution of the first time, τx, a Lévy process Xt crosses a fixed negative level -x. We prove that ℙ{τx &gt;t} ~V(x) ℙ{Xt≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τx&gt;t} explicitly in both light- and heavy-tailed cases.</jats:p