scholarly journals Rare Events and Conditional Events on Random Strings

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Mireille Régnier ◽  
Alain Denise
Keyword(s):  
Large Deviation ◽  
Probability Model ◽  
Rare Event ◽  
Bernoulli Model ◽  
Single Word ◽  
Tail Distribution ◽  
A Genome ◽  
International Audience ◽  
Conditional Events ◽  
Sharp Large Deviation

International audience Some strings -the texts- are assumed to be randomly generated, according to a probability model that is either a Bernoulli model or a Markov model. A rare event is the over or under-representation of a word or a set of words. The aim of this paper is twofold. First, a single word is given. One studies the tail distribution of the number of its occurrences. Sharp large deviation estimates are derived. Second, one assumes that a given word is overrepresented. The distribution of a second word is studied; formulae for the expectation and the variance are derived. In both cases, the formulae are accurate and actually computable. These results have applications in computational biology, where a genome is viewed as a text.

A conditional limit theorem for high-dimensional ℓᵖ-spheres

2018 ◽  
Vol 55 (4) ◽  
pp. 1060-1077 ◽  
Author(s):  
Steven S. Kim ◽  
Kavita Ramanan
Keyword(s):  
Limit Theorem ◽  
Large Deviation ◽  
Convex Geometry ◽  
Rare Event ◽  
High Dimensional ◽  
Random Projections ◽  
Dimensional Euclidean Space ◽  
High Dimensions ◽  
Geometric Setting ◽  
Conditional Limit

Abstract The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

State-dependent importance sampling for regularly varying random walks

2008 ◽  
Vol 40 (04) ◽  
pp. 1104-1128 ◽  
Author(s):  
Jose H. Blanchet ◽  
Jingchen Liu
Keyword(s):  
Importance Sampling ◽  
Large Deviation ◽  
Rare Event ◽  
Parametric Family ◽  
Finite Variance ◽  
Or Efficiency ◽  
State Dependent ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Path Dependent

Consider a sequence (X k : k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (S n : n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of S n in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

scholarly journals Computation of extreme heat waves in climate models using a large deviation algorithm

2017 ◽  
Vol 115 (1) ◽  
pp. 24-29 ◽  
Author(s):  
Francesco Ragone ◽  
Jeroen Wouters ◽  
Freddy Bouchet
Keyword(s):  
Statistical Physics ◽  
Climate Models ◽  
Large Deviation ◽  
Heat Waves ◽  
Rare Event ◽  
Teleconnection Pattern ◽  
Extreme Heat ◽  
Sampling Efficiency ◽  
Wide Range ◽  
The Impact

Studying extreme events and how they evolve in a changing climate is one of the most important current scientific challenges. Starting from complex climate models, a key difficulty is to be able to run long enough simulations to observe those extremely rare events. In physics, chemistry, and biology, rare event algorithms have recently been developed to compute probabilities of events that cannot be observed in direct numerical simulations. Here we propose such an algorithm, specifically designed for extreme heat or cold waves, based on statistical physics. This approach gives an improvement of more than two orders of magnitude in the sampling efficiency. We describe the dynamics of events that would not be observed otherwise. We show that European extreme heat waves are related to a global teleconnection pattern involving North America and Asia. This tool opens up a wide range of possible studies to quantitatively assess the impact of climate change.

Rare events in series of queues

1992 ◽  
Vol 29 (1) ◽  
pp. 168-175 ◽  
Author(s):  
Pantelis Tsoucas
Keyword(s):  
Queueing Networks ◽  
Total Population ◽  
Large Deviation ◽  
Rare Event ◽  
Arrival Rate ◽  
Service Rate ◽  
Action Functional ◽  
In Series ◽  
Service Rates ◽  
Queues In Series

In an ergodic network of K M/M/1 queues in series we consider the rare event that, as N increases, the total population in the network exceeds N during a busy period. By utilizing the contraction principle of large deviation theory, an action functional is obtained for this exit problem. The ensuing minimization is carried out for K = 2 and an indication is given for arbitrary K. It is shown that, asymptotically and for unequal service rates, the ‘most likely' path for this rare event is one where the arrival rate has been interchanged with the smallest service rate. The problem has been posed in Parekh and Walrand [7] in connection with importance sampling simulation methods for queueing networks. Its solution has previously been obtained only heuristically.

scholarly journals State-dependent importance sampling for regularly varying random walks

2008 ◽  
Vol 40 (4) ◽  
pp. 1104-1128 ◽  
Author(s):  
Jose H. Blanchet ◽  
Jingchen Liu
Keyword(s):  
Importance Sampling ◽  
Large Deviation ◽  
Rare Event ◽  
Parametric Family ◽  
Finite Variance ◽  
Or Efficiency ◽  
State Dependent ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Path Dependent

Consider a sequence (Xk:k≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (Sn:n≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail ofSnin a large deviation regime asn↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

scholarly journals The height of the Lyndon tree

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Lucas Mercier ◽  
Philippe Chassaing
Keyword(s):  
Binary Tree ◽  
Large Deviation ◽  
Random Variables ◽  
Convergence In Probability ◽  
Eulerian Numbers ◽  
Deviation Rate ◽  
Rate Functions ◽  
International Audience ◽  
Lyndon Words

International audience We consider the set $\mathcal{L}_n<$ of n-letters long Lyndon words on the alphabet $\mathcal{A}=\{0,1\}$. For a random uniform element ${L_n}$ of the set $\mathcal{L}_n$, the binary tree $\mathfrak{L} (L_n)$ obtained by successive standard factorization of $L_n$ and of the factors produced by these factorization is the $\textit{Lyndon tree}$ of $L_n$. We prove that the height $H_n$ of $\mathfrak{L} (L_n)$ satisfies $\lim \limits_n \frac{H_n}{\mathsf{ln}n}=\Delta$, in which the constant $\Delta$ is solution of an equation involving large deviation rate functions related to the asymptotics of Eulerian numbers ($\Delta ≃5.092\dots $). The convergence is the convergence in probability of random variables.

scholarly journals Efficient Simulation of Large Deviation Events for Sums of Random Vectors Using Saddle-Point Representations

2013 ◽  
Vol 50 (3) ◽  
pp. 703-720
Author(s):  
Ankush Agarwal ◽  
Santanu Dey ◽  
Sandeep Juneja
Keyword(s):  
Saddle Point ◽  
Importance Sampling ◽  
Relative Error ◽  
Large Deviation ◽  
Rare Event ◽  
Characteristic Functions ◽  
Error Estimator ◽  
Risk Modeling ◽  
Random Vectors ◽  
Efficient Simulation

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 queueing 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.

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