We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in DeanDup09 that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffice to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n^{2β+1}) function evaluations suffice to achieve a given relative precision, where β is the number of bottleneck stations in the network. This is the first rigorous analysis that allows to favorably compare splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n^{d}) variables.
This paper studies the queue length process in series Jackson networks with external input to the first station. We show that its Markov transition probabilities can be written as a finite sum of non-crossing probabilities, so that questions on time-dependent queueing behavior are translated to questions on non-crossing probabilities. This makes previous work on non-crossing probabilities relevant to queueing systems and allows new queueing results to be established. To illustrate the latter, we prove that the relaxation time (i.e., the reciprocal of the `spectral gap') of a positive recurrent…
Importance sampling (IS) is a variance reduction method for simulating rare events. A recent paper by Dupuis, Wang and Sezer (Ann. App. Probab. 17(4):1306- 1346, 2007) exploits connections between IS and stochastic games and optimal control problems to show how to design and analyze simple and efficient IS algorithms for various overflow events for tandem Jackson networks. The present paper uses the same approach to build asymptotically optimal IS schemes for stable open Jackson networks with a tree topology. Customers arrive at the single root of the tree. The rare overflow event we consider …
We study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network (GJN for short), assuming its stability. For the two station case, this problem has been recently solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probab…
We develop a new algorithm for the estimation of rare event probabilities associated with the steady-state of a Markov stochastic process with continuous state space $\mathbb R^d$ and discrete time steps (i.e. a discrete-time $\mathbb R^d$-valued Markov chain). The algorithm, which we coin Recurrent Multilevel Splitting (RMS), relies on the Markov chain's underlying recurrent structure, in combination with the Multilevel Splitting method. Extensive simulation experiments are performed, including experiments with a nonlinear stochastic model that has some characteristics of complex climate mode…
In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology a Markov chain is simulated, with that conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a h…
Improving Importance Sampling estimators for rare event probabilities requires sharp approximations of conditional densities. This is achieved for events E_{n}:=(f(X_{1})+...+f(X_{n}))\inA_{n} where the summands are i.i.d. and E_{n} is a large or moderate deviation event. The approximation of the conditional density of the real r.v's X_{i} 's, for 1\leqi\leqk_{n} with repect to E_{n} on long runs, when k_{n}/n\to1, is handled. The maximal value of k compatible with a given accuracy is discussed; algorithms and simulated results are presented.
We analyse the splitting algorithm performance in the estimation of rare event probabilities and this in a discrete multidimensional framework. For this we assume that each threshold is partitioned into disjoint subsets and the probability for a particle to reach the next threshold will depend on the starting subset. A straightforward estimator of the rare event probability is given by the proportion of simulated particles for which the rare event occurs. The variance of this estimator we get is the sum of two parts: one part resuming the variability due to each threshold and a second part res…
We consider the general problem of estimating probabilities which arise as a union of dependent events. We propose a flexible series of estimators for such probabilities, and describe variance reduction schemes applied to the proposed estimators. We derive efficiency results of the estimators in rare-event settings, in particular those associated with extremes. Finally, we examine the performance of our estimators in a numerical example.
In this paper we consider the extreme behavior of the extremal eigenvalues of white Wishart matrices, which plays an important role in multivariate analysis. In particular, we focus on the case when the dimension of the feature p is much larger than or comparable to the number of observations n, a common situation in modern data analysis. We provide asymptotic approximations and bounds for the tail probabilities of the extremal eigenvalues. Moreover, we construct efficient Monte Carlo simulation algorithms to compute the tail probabilities. Simulation results show that our method has the best …
This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simula…
Efficient Rare-Event Simulation for Random Geometric Graphs via Importance Sampling
School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Data Science and Statistics, Aarhus University（奥胡斯大学数据科学与统计学）；Institute of Stochastics, Ulm University（乌尔姆大学随机过程研究所）；School of Mathematics and Physics, The University of Queensland（昆士兰大学数学与物理学院）
This paper introduces a new Importance Sampling scheme, called Adaptive Twisted Importance Sampling, which is adequate for the improved estimation of rare event probabilities in he range of moderate deviations pertaining to the empirical mean of real i.i.d. summands. It is based on a sharp approximation of the density of long runs extracted from a random walk conditioned on its end value.
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 $E(Γ(f) | \sup_T f(t) > b)$ 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 Hölder continuo…
The event of large losses plays an important role in credit risk. As these large losses are typically rare, and portfolios usually consist of a large number of positions, large deviation theory is the natural tool to analyze the tail asymptotics of the probabilities involved. We first derive a sample-path large deviation principle (LDP) for the portfolio's loss process, which enables the computation of the logarithmic decay rate of the probabilities of interest. In addition, we derive exact asymptotic results for a number of specific rare-event probabilities, such as the probability of the los…
In the framework of dependent risks it is a crucial task for risk management purposes to quantify the probability that the aggregated risk exceeds some large value u. Motivated by Asmussen et al. (2011) in this paper we introduce a modified Asmussen-Kroese estimator for simulation of the rare event that the aggregated risk exceeds u. We show that in the framework of log-Gaussian risks our novel estimator has the best possible performance. For the more general class of log-elliptical risks with marginal distributions in the Gumbel max-domain of attraction we propose a modified Rojas-Nandayapa e…
In this work, we propose an algorithm to simulate rare events for electronic circuit design. Our approach heavily relies on a smart use of importance sampling, which enables us to tackle probabilities of the magnitude 10 --10. Not only can we compute very rare default probability, but we can also compute the quantile associated to a given default probability and its expected shortfall. We show the impressive efficiency of method on real circuits.
We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities within a regenerative cycle. Exponential twisting based state-independent methods, which are effective in efficiently estimating these probabilities for light-tailed increments are not applicable when the increments are heavy-tailed. To address the latter case, more complex and …
Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a priori fixed change of measure suggested by large deviation analysis, has been shown to fail in even the simplest network setting (e.g., a two-node tandem network). Exploiting connections between importance sampling, differential games, and classical subsolutions of the corresponding Isaacs equation, we…
We study probabilities of various rare events for the limiting point process that appears at the random matrix hard edge. We also show a transition from hard edge to bulk behavior. Asymptotic events studied include a central limit theorem and large deviation result for the number of points in a growing interval $[0,λ]$ as $λ\to \infty$. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the Sine$_β$ process.
Let $X$ be the constrained random walk on ${\mathbb Z}_+^d$ representing the queue lengths of a stable Jackson network and $x$ its initial position. Let $τ_n$ be the first time the sum of the components of $X$ equals $n$. $p_n \doteq P_x(τ_n < τ_0)$ is a key performance measure for the queueing system represented by $X$, stability implies $p_n\rightarrow 0$ exponentially. Currently the only analytic method available to approximate $p_n$ is large deviations analysis, which gives the exponential decay rate of $p_n$. Finer results are available via rare event simulation. The present article devel…
This paper considers the problem of simultaneously estimating rare-event probabilities for a class of Gaussian random fields. A conventional rare-event simulation method is usually tailored to a specific rare event and consequently would lose estimation efficiency for different events of interest, which often results in additional computational cost in such simultaneous estimation problem. To overcome this issue, we propose a uniformly efficient estimator for a general family of Hölder continuous Gaussian random fields. We establish the asymptotic and uniform efficiency of the proposed method …
We study small white noise perturbations of planar dynamical systems with heteroclinic networks in the limit of vanishing noise. We show that the probabilities of transitions between various cells that the network tessellates the plane into decay as powers of the noise magnitude. We show that the most likely scenario for the realization of these rare transition events involves spending atypically long times in the neighborhoods of certain saddle points of the network. We describe the hierarchy of time scales and clusters of accessibility associated with these rare transition events. We discuss…
In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable $V$ satisfying the distributional equation $V\stackrel{\mathcal{D}}{=}f(V)$, where $f(v)=A\max\{v,D\}+B$ for $(A,B,D)\in(0,\infty)\times {\mathbb{R}}^2$. This paper is concerned with computational methods for evaluating these tail probabilities. We introduce a novel importance sampling algorithm, involving an exponential shift over a random time interval, for estimating these rare event probabilities. We prove that the proposed estimator…
We consider systems of stochastic differential equations with multiple scales and small noise and assume that the coefficients of the equations are ergodic and stationary random fields. Our goal is to construct provably-efficient importance sampling Monte Carlo methods that allow efficient computation of rare event probabilities or expectations of functionals that can be associated with rare events. Standard Monte Carlo algorithms perform poorly in the small noise limit and hence fast simulations algorithms become relevant. The presence of multiple scales complicates the design and the analysi…
Multilevel Splitting methods, also called Sequential Monte-Carlo or \emph{Subset Simulation}, are widely used methods for estimating extreme probabilities of the form $P[S(\mathbf{U}) > q]$ where $S$ is a deterministic real-valued function and $\mathbf{U}$ can be a random finite- or infinite-dimensional vector. Very often, $X := S(\mathbf{U})$ is supposed to be a continuous random variable and a lot of theoretical results on the statistical behaviour of the estimator are now derived with this hypothesis. However, as soon as some threshold effect appears in $S$ and/or $\mathbf{U}$ is discrete o…
We propose a new method for estimating rare event probabilities when independent samples are available. It is assumed that the underlying probability measures satisfy a large deviations principle with a scaling parameter $\varepsilon$ that we call temperature. We show how by combining samples at different temperatures, one can construct an estimator with greatly reduced variance. Although as presented here the method is not as broadly applicable as other rare event simulation methods, such as splitting or importance sampling, it does not require any problem-dependent constructions.
In this work, we propose a class of importance sampling (IS) estimators for estimating the right tail probability of a sum of continuous random variables based on a change of variables to $L^1$ polar coordinates in which the radial and angular components of the IS distribution are considered separately. When the asymptotic behaviour of the sum is known we exploit it for the radial change of measure, and the resulting estimator has the appealing form of the (known) asymptotic multiplied by a random multiplicative correction factor. Given we assume knowledge of the asymptotic behaviour of the su…
The Adaptive Multilevel Splitting algorithm is a very powerful and versatile method to estimate rare events probabilities. It is an iterative procedure on an interacting particle system, where at each step, the $k$ less well-adapted particles among $n$ are killed while $k$ new better adapted particles are resampled according to a conditional law. We analyze the algorithm in the idealized setting of an exact resampling and prove that the estimator of the rare event probability is unbiased whatever $k$. We also obtain a precise asymptotic expansion for the variance of the estimator and the cost …
Much work in the study of large deviations for random graph models is focused on the dense regime where the theory of graphons has emerged as a principal tool. These tools do not give a good approach to large deviation problems for random graph models in the sparse regime. The aim of this paper is to study an approach for large deviation problems in this regime by establishing Large Deviation Principles (LDP) on suitable path spaces for certain exploration processes of the associated random graph sequence. Our work focuses on the study of one particular class of random graph models, namely the…
We present a probabilistic T-cell model that includes negative selection and takes contrasting models of tissue-restricted antigen (TRA) expression in the thymus into account. We start from the basic model of van den Berg, Rand, and Burroughs (2001) and include negative selection via individual-based T-cell modelling, in which each T-cell is defined by its stimulation rates to all relevant self antigens. We present a simulation approach based on partial tilting of the stimulation rates recognized by a single T-cell. We investigate the effects of negative selection for diverging modes of thymic…
In dense Erdős-Rényi random graphs, we are interested in the events where large numbers of a given subgraph occur. The mean behavior of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is natural to ask, can one develop efficient numerical schemes to estimate the probability of an Erdős-Rényi graph containing an excessively large number of a fixed given subgraph? Using the large deviation principle we study an importance sampling scheme as a method to numerically compute the small probabilities of large triangle counts occurring…
