Efficient Simulation and Conditional Functional Limit Theorems for Ruinous Heavy-tailed Random Walks

The contribution of this paper is to introduce change of measure based techniques for the rare-event analysis of heavy-tailed stochastic processes. Our changes-of-measure are parameterized by a family of distributions admitting a mixture form. We exploit our methodology to achieve two types of results. First, we construct Monte Carlo estimators that are strongly efficient (i.e. have bounded relative mean squared error as the event of interest becomes rare). These estimators are used to estimate both rare-event probabilities of interest and associated conditional expectations. We emphasize that our techniques allow us to control the expected termination time of the Monte Carlo algorithm even if the conditional expected stopping time (under the original distribution) given the event of interest is infinity -- a situation that sometimes occurs in heavy-tailed settings. Second, the mixture family serves as a good approximation (in total variation) of the conditional distribution of the whole process given the rare event of interest. The convenient form of the mixture family allows us to obtain, as a corollary, functional conditional central limit theorems that extend classical results in the literature. We illustrate our methodology in the context of the ruin probability $P(\sup_n S_n >b)$, where $S_n$ is a random walk with heavy-tailed increments that have negative drift. Our techniques are based on the use of Lyapunov inequalities for variance control and termination time. The conditional limit theorems combine the application of Lyapunov bounds with coupling arguments

Efficient simulation of large deviation events for sums of random vectors using saddle-point representations

We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for a sum of independent, identically distributed (i.i.d.), light-tailed and nonlattice random vectors. The latter problem besides being of independent interest, also forms a building block for more complex rare event problems that arise, for instance, in queuing and financial credit risk modeling. It has been extensively studied in the literature where state-independent, exponential-twisting-based importance sampling has been shown to be asymptotically efficient and a more nuanced state-dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point-based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. These representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Furthermore, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and show that they possess the asymptotically vanishing relative error property that is stronger than the bounded relative error property. To illustrate the broader applicability of the proposed methodology, we extend it to develop an asymptotically vanishing relative error estimator for the practically important expected overshoot of sums of i.i.d. random variables

On the Sum of Order Statistics and Applications to Wireless Communication Systems Performances

We consider the problem of evaluating the cumulative distribution function (CDF) of the sum of order statistics, which serves to compute outage probability (OP) values at the output of generalized selection combining receivers. Generally, closed-form expressions of the CDF of the sum of order statistics are unavailable for many practical distributions. Moreover, the naive Monte Carlo (MC) method requires a substantial computational effort when the probability of interest is sufficiently small. In the region of small OP values, we propose instead two effective variance reduction techniques that yield a reliable estimate of the CDF with small computing cost. The first estimator, which can be viewed as an importance sampling estimator, has bounded relative error under a certain assumption that is shown to hold for most of the challenging distributions. An improvement of this estimator is then proposed for the Pareto and the Weibull cases. The second is a conditional MC estimator that achieves the bounded relative error property for the Generalized Gamma case and the logarithmic efficiency in the Log-normal case. Finally, the efficiency of these estimators is compared via various numerical experiments

Efficient rare-event simulation for the maximum of heavy-tailed random walks

Let $(X_n:n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set $S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling algorithm to estimate the tail of $M=\max \{S_n:n\geq 0\}$ that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund's (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunov-type inequalities that can be useful in the study of more general importance sampling algorithms.Comment: Published in at http://dx.doi.org/10.1214/07-AAP485 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Efficient simulation of density and probability of large deviations of sum of random vectors using saddle point representations

We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for a sum of independent, identically distributed, light-tailed and non-lattice random vectors. The latter problem besides being of independent interest, also forms a building block for more complex rare event problems that arise, for instance, in queueing and financial credit risk modelling. It has been extensively studied in literature where state independent exponential twisting based importance sampling has been shown to be asymptotically efficient and a more nuanced state dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. These representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Further, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and show that they possess the asymptotically vanishing relative error property that is stronger than the bounded relative error property. To illustrate the broader applicability of the proposed methodology, we extend it to similarly efficiently estimate the practically important expected overshoot of sums of iid random variables