Tail approximations of integrals of Gaussian random fields

This paper develops asymptotic approximations of $P(\int_Te^{f(t)}\,dt>b)$ as $b\rightarrow\infty$ for a homogeneous smooth Gaussian random field, $f$, living on a compact $d$-dimensional Jordan measurable set $T$. The integral of an exponent of a Gaussian random field is an important random variable for many generic models in spatial point processes, portfolio risk analysis, asset pricing and so forth. The analysis technique consists of two steps: 1. evaluate the tail probability $P(\int_{\Xi}e^{f(t)}\,dt>b)$ over a small domain $\Xi$ depending on $b$, where $\operatorname {mes}(\Xi)\rightarrow0$ as $b\rightarrow \infty$ and $\operatorname {mes}(\cdot)$ is the Lebesgue measure; 2. with $\Xi$ appropriately chosen, we show that $P(\int_Te^{f(t)}\,dt>b)=(1+o(1))\operatorname{mes}(T)\times \operatorname{mes}^{-1}(\Xi)P(\int_{\Xi}e^{f(t)}\,dt>b)$.Comment: Published in at http://dx.doi.org/10.1214/10-AOP639 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Extreme Analysis of a Non-convex and Nonlinear Functional of Gaussian Processes -- On the Tail Asymptotics of Random Ordinary Differential Equations

In this paper, we consider a stochastic system described by a differential equation admitting a spatially varying random coefficient. The differential equation has been employed to model various static physics systems such as elastic deformation, water flow, electric-magnetic fields, temperature distribution, etc. A random coefficient is introduced to account for the system's uncertainty and/or imperfect measurements. This random coefficient is described by a Gaussian process (the input process) and thus the solution to the differential equation (under certain boundary conditions) is a complexed functional of the input Gaussian process. In this paper, we focus the analysis on the one-dimensional case and derive asymptotic approximations of the tail probabilities of the solution to the equation that has various physics interpretations under different contexts. This analysis rests on the literature of the extreme analysis of Gaussian processes (such as the tail approximations of the supremum) and extends the analysis to more complexed functionals.Comment: supplementary material is include

Rare-event Simulation and Efficient Discretization for the Supremum of Gaussian Random Fields

In this paper, we consider a classic problem concerning the high excursion probabilities of a Gaussian random field $f$ living on a compact set $T$. We develop efficient computational methods for the tail probabilities $P(\sup_T f(t) > b)$ and the conditional expectations $E(\Gamma(f) | \sup_T f(t) > b)$ as $b\rightarrow \infty$. For each $\varepsilon$ positive, we present Monte Carlo algorithms that run in \emph{constant} time and compute the interesting quantities with $\varepsilon$ relative error for arbitrarily large $b$. The efficiency results are applicable to a large class of H\"older continuous Gaussian random fields. Besides computations, the proposed change of measure and its analysis techniques have several theoretical and practical indications in the asymptotic analysis of extremes of Gaussian random fields

On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields

In this paper, we consider the extreme behavior of a Gaussian random field $f(t)$ living on a compact set $T$. In particular, we are interested in tail events associated with the integral $\int_Te^{f(t)}\,dt$. We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field $f$ (given that $\int_Te^{f(t)}\,dt$ exceeds a large value) in total variation. Based on this approximation, we show that the tail event of $\int_Te^{f(t)}\,dt$ is asymptotically equivalent to the tail event of $\sup_T\gamma(t)$ where $\gamma(t)$ is a Gaussian process and it is an affine function of $f(t)$ and its derivative field. In addition to the asymptotic description of the conditional field, we construct an efficient Monte Carlo estimator that runs in polynomial time of $\log b$ to compute the probability $P(\int_Te^{f(t)}\,dt>b)$ with a prescribed relative accuracy.Comment: Published in at http://dx.doi.org/10.1214/13-AAP960 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Discussion of: A statistical analysis of multiple temperature proxies: Are reconstructions of surface temperatures over the last 1000 years reliable?

Discussion of "A statistical analysis of multiple temperature proxies: Are reconstructions of surface temperatures over the last 1000 years reliable?" by B.B. McShane and A.J. Wyner [arXiv:1104.4002]Comment: Published in at http://dx.doi.org/10.1214/10-AOAS398C the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org