Large Deviations Techniques Research Articles

Half-space depth of log-concave probability measures

Given a probability measure mu on {{mathbb {R}}}^n, Tukey’s half-space depth is defined for any xin {{mathbb {R}}}^n by varphi _{mu }(x)=inf {mu (H):Hin {{{mathcal {H}}}}(x)}, where mathcal{H}(x) is the set of all half-spaces H of {{mathbb {R}}}^n containing x. We show that if mu is a non-degenerate log-concave probability measure on {{mathbb {R}}}^n then e-c1n⩽∫Rnφμ(x)dμ(x)⩽e-c2n/Lμ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} e^{-c_1n}\\leqslant \\int _{{\\mathbb {R}}^n}\\varphi _{\\mu }(x)\\,d\\mu (x) \\leqslant e^{-c_2n/L_{\\mu }^2} \\end{aligned}$$\\end{document}where L_{mu } is the isotropic constant of mu and c_1,c_2>0 are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of L_q-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.

The Randomised Heston Model

We propose a randomised version of the Heston model -- a widely used stochastic volatility model in mathematical finance -- assuming that the starting point of the variance process is a random variable. In such a system, we study the small- and large-time behaviours of the implied volatility, and show that the proposed randomisation generates a short-maturity smile much steeper ('with explosion') than in the standard Heston model, thereby palliating the deficiency of classical stochastic volatility models in short time. We precisely quantify the speed of explosion of the smile for short maturities in terms of the right tail of the initial distribution, and in particular show that an explosion rate of t^r (r in [0,1/2]) for the squared implied volatility -- as observed on market data -- can be obtained by a suitable choice of randomisation. The proofs are based on large deviations techniques and the theory of regular variations.

Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains.

We consider the maximum entropy Markov chain inference approach to characterize the collective statistics of neuronal spike trains, focusing on the statistical properties of the inferred model. To find the maximum entropy Markov chain, we use the thermodynamic formalism, which provides insightful connections with statistical physics and thermodynamics from which large deviations properties arise naturally. We provide an accessible introduction to the maximum entropy Markov chain inference problem and large deviations theory to the community of computational neuroscience, avoiding some technicalities while preserving the core ideas and intuitions. We review large deviations techniques useful in spike train statistics to describe properties of accuracy and convergence in terms of sampling size. We use these results to study the statistical fluctuation of correlations, distinguishability, and irreversibility of maximum entropy Markov chains. We illustrate these applications using simple examples where the large deviation rate function is explicitly obtained for maximum entropy models of relevance in this field.

Concentration for Coulomb gases and Coulomb transport inequalities

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann–Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. Our work is inspired by the one of Maïda and Maurel-Segala, itself inspired by large deviations techniques. Our approach allows to recover, extend, and simplify previous results by Rougerie and Serfaty.

Distribution of diameters for Erdős-Rényi random graphs.

We study the distribution of diameters d of Erdős-Rényi random graphs with average connectivity c. The diameter d is the maximum among all the shortest distances between pairs of nodes in a graph and an important quantity for all dynamic processes taking place on graphs. Here we study the distribution P(d) numerically for various values of c, in the nonpercolating and percolating regimes. Using large-deviation techniques, we are able to reach small probabilities like 10^{-100} which allow us to obtain the distribution over basically the full range of the support, for graphs up to N=1000 nodes. For values c<1, our results are in good agreement with analytical results, proving the reliability of our numerical approach. For c>1 the distribution is more complex and no complete analytical results are available. For this parameter range, P(d) exhibits an inflection point, which we found to be related to a structural change of the graphs. For all values of c, we determined the finite-size rate function Φ(d/N) and were able to extrapolate numerically to N→∞, indicating that the large-deviation principle holds.

The asymptotic smile of a multiscaling stochastic volatility model

We consider a stochastic volatility model which captures relevant stylized facts of financial series, including the multi-scaling of moments. The volatility evolves according to a generalized Ornstein–Uhlenbeck processes with super-linear mean reversion.Using large deviations techniques, we determine the asymptotic shape of the implied volatility surface in any regime of small maturity t→0 or extreme log-strike |κ|→∞ (with bounded maturity). Even if the price has continuous paths, out-of-the-money implied volatility diverges for small maturity, producing a very pronounced smile.

On the probability of current and temperature overloading in power grids

The advent of renewable energy sources has huge implications for the design and control of power grids. On the engineering side, reliability is currently ensured by strict con- straints on current, voltage and temperature. However, with growing supply-side uncertainty induced by renewables, these will need to be replaced by probabilistic guarantees, allowing constraints on a given line to be violated with a low probability, e.g., several minutes per year. In the present note we illustrate, using large deviations techniques, how replacing (probabilistic) current constraints by temperature constraints can lead to capacity gains in power grids.

Tail approximations in credit portfolios using large deviations techniques

In this paper, it is analyzed how Large Deviations (LD) techniques can be used for practical credit portfolio management. Applications include the internal risk management for a large credit portfolio, or the overall need to meet external requirements imposed by Basel II. For this purpose the paper provides fast and reliable methods for the computation of Value at Risk (VaR) and Conditional Value at Risk (CVaR) in general factor models using LD. Recovery rate (RR) is modelled as a random variable, depending on the state of the economy, using different models from the literature. The applicability of LD is shown, above mentioned risk measures are calculated and compared to the case of deterministic RR, demonstrating the need of an adequate modelling of RRs. Optimal portfolios with regard to either VaR or CVaR under portfolio constraints are derived, showing that LD techniques can outperform Monte Carlo (MC) simulations with regard to computational effort without a significant lack of accuracy.

Asymptotics of Forward Implied Volatility

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Levy models. This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques and allows us to recover (in particular) many results for the spot implied volatility smile. In passing we (i) show that the forward-start date has to be rescaled in order to obtain nontrivial small-maturity asymptotics, (ii) prove that the forward-start date may influence the large-maturity behavior of the forward smile, and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.

Asymptotics of Forward Implied Volatility

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Levy models.This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile.In passing we (i) show that the forward-start date has to be rescaled in order to obtain non-trivial small-maturity asymptotics,(ii) prove that the forward-start date may influence the large-maturity behaviour of the forward smile,and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.

State-dependent importance sampling for a slowdown tandem queue

In this paper we investigate an advanced variant of the classical (Jackson) tandem queue, viz. a two-node system with server slowdown. By this mechanism, the service speed of the upstream queue is reduced as soon as the number of jobs in the downstream queue reaches some pre-specified threshold. We focus on the estimation of the probability of overflow in the downstream queue before the system becomes empty, starting from any given state in the state space. The principal contribution of this paper is that we construct importance sampling schemes to estimate these probabilities in case they are small; in particular: (1)We use powerful heuristics to identify the exponential decay rate of the probability under consideration, and verify this result by applying sample-path large deviations techniques. (2) Based on these heuristics we develop a change of measure to be used in importance sampling. (3) We prove that this scheme is asymptotically efficient, using a shorter and more straightforward method than usually provided in the literature. Unfortunately, this scheme is difficult to use in practice, therefore (4) we propose an algorithm that offers considerable computational advantage over the first scheme. For this scheme we provide a proof of asymptotic efficiency for certain parameter settings, as well as numerical results showing that the scheme works well for all parameters.

A Backlog Evaluation Formula for Admission Control Based on the Stochastic Network Calculus with Many Flows

Admission control is a procedure to guarantee a given level of Quality of Service (QoS) by accepting or rejecting arrival connection requests. There are many studies on backlog or loss rate evaluation formulas for admission control at a single node. However, there are few studies on end-to-end evaluation formulas suitable for admission control. In a previous paper, the authors proposed a new stochastic network calculus for many flows using an approach taken from large deviations techniques and obtained asymptotic end-to-end evaluation formulas for output burstiness and backlog. In this paper, we apply this stochastic network calculus to a heterogeneous tandem network with many forwarding flows and cross traffic flows constrained by leaky buckets, and obtain a simple evaluation formula for the end-to-end backlog. In this formula, the end-to-end backlog can be evaluated by the traffic load at the bottle neck node. This result leads us to a natural extension of the evaluation formula for a single node.

Statistical anomaly detection with sensor networks

We seek to detect statistically significant temporal or spatial changes in either the underlying process the sensor network is monitoring or in the network operation itself. These changes may point to faults, adversarial threats, misbehavior, or other anomalies that require intervention. To that end, we introduce a new statistical anomaly detection framework that uses Markov models to characterize the “normal” behavior of the sensor network. We develop a series of Markov models, including tree-indexed Markov chains which can model its spatial structure. For each model, an anomaly-free probability law is estimated from past traces. We leverage large deviations techniques to develop optimal anomaly detection rules for each corresponding Markov model, assessing whether its most recent empirical measure is consistent with the anomaly-free probability law. A series of simulation results, some with real sensor data, validate the effectiveness of the proposed anomaly detection algorithms.

Optimal aggregation of noisy observations: A large deviations approach

Sensing and data aggregation tasks in distributed systems should not be considered as separate issues. The quality of collective estimation involves a fundamental tradeoff between sensing quality, which can be increased by increasing the number of sensors, and aggregation quality under a given capacity of the network, which decreases if the number of sensors is too large. In this paper, we examine a system level strategy for optimal aggregation of data from an ensemble of independent sensors. In particular, we consider large scale aggregation from very many sensors, in which case the network capacity diverges to infinity. Then, by applying the large deviations techniques, we conclude the following significant result: larger scale aggregation always outperforms smaller scale aggregation at higher noise levels, while below a critical value of noise, there exist moderate scale aggregation levels at which optimal estimation is realized. At a critical value of noise, there is an abrupt change in the behavior of a parameter characterizing the aggregation strategy, similar to a phase transition in statistical physics.

A characterization of dimension free concentration in terms of transportation inequalities

The aim of this paper is to give a characterization of the dimension free concentration of measure phenomenon in terms of transportation-cost inequalities. We apply this theorem to give a new and very short proof of a result by Otto and Villani. Another application is to show that the Poincaré inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of all these results rely on simple Large Deviations techniques.

Assessing the relative power of structural break tests using a framework based on the approximate Bahadur slope

Abstract We compare the asymptotic relative efficiency of the Exp, Mean, and Sup functionals of the Wald, LM and LR tests for structural change analyzed by Andrews [Andrews, D.W.K., 1993. Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856.] and Andrews and Ploberger [Andrews, D.W.K., Ploberger, W., 1994. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414.]. We derive the approximate Bahadur slopes of these tests using large deviations techniques. These show that tests based on the Mean functional are inferior to those based on the Sup and Exp when using the same base statistic. Also, for a given functional, the Wald-based test dominates the LR-based test, which dominates the LM-based one. We show that the Sup- and Mean-type tests satisfy Wieand’s [Wieand, H.S., 1976. A Condition under which the Pitman and Bahadur approaches to efficiency coincide. Annals of Statistics 4, 1003–1011.] condition so that their slopes yield the limiting (as the size tends to zero) asymptotic relative Pitman efficiency (whether this holds for the Exp-type tests still remains a conjecture). Using this measure of efficiency, the Mean-type tests are also inferior to the Sup. We also compare tests based on the Wald and LM statistics modified with a HAC estimator. In this case, the inferiority of the LM-based tests is especially pronounced. The relevance of our theoretical results in finite samples is assessed via simulations. Our results are in contrast to those obtained by the authors–in the second reference cited above–based on a local asymptotic framework, and our analysis thereby reveals its potential weaknesses in the context of structural change problems.

On the delay performance in multi-antenna wireless networks using contention-based feedback

When a spread spectrum contention-based feedback channel is employed, it has been shown that most of the multiuser diversity gain can be maintained without any delay constraint. To address the delay in sending packets, we investigate how a spread spectrum contention-based feedback channel affects the delay and throughput performance. Using large deviations techniques, we show that most of the multiuser diversity gain can be maintained while satisfying a delay constraint in which the probability that the maximum delay of any bits in any users' queues is less than a specified value. In addition, we show that the spectral efficiency improves as the number of users increases while maintaining a fixed normalized maximum delay (which is defined as the ratio of the maximum delay and the number of users) as the number of users increases.

A note on convergence rates for posterior distributions via large deviations techniques

This paper provides rates of convergence for a class of posterior distributions which are exchangeable with respect to a tree of partitions. The derivations rely on large deviations techniques.

Computable exponential bounds for screened estimation and simulation

Suppose the expectation $E(F(X))$ is to be estimated by the empirical averages of the values of $F$ on independent and identically distributed samples $\{X_i\}$. A sampling rule called the "screened" estimator is introduced, and its performance is studied. When the mean $E(U(X))$ of a different function $U$ is known, the estimates are "screened," in that we only consider those which correspond to times when the empirical average of the $\{U(X_i)\}$ is sufficiently close to its known mean. As long as $U$ dominates $F$ appropriately, the screened estimates admit exponential error bounds, even when $F(X)$ is heavy-tailed. The main results are several nonasymptotic, explicit exponential bounds for the screened estimates. A geometric interpretation, in the spirit of Sanov's theorem, is given for the fact that the screened estimates always admit exponential error bounds, even if the standard estimates do not. And when they do, the screened estimates' error probability has a significantly better exponent. This implies that screening can be interpreted as a variance reduction technique. Our main mathematical tools come from large deviations techniques. The results are illustrated by a detailed simulation example.

Random Quantum Codes from Gaussian Ensembles and an Uncertainty Relation

Using random Gaussian vectors and an information-uncertainty relation, we give a proof that the coherent information is an achievable rate for entanglement transmission through a noisy quantum channel. The codes are random subspaces selected according to the Haar measure, but distorted as a function of the sender's input density operator. Using large deviations techniques, we show that classical data transmitted in either of two Fourier-conjugate bases for the coding subspace can be decoded with low probability of error. A recently discovered information-uncertainty relation then implies that the quantum mutual information for entanglement encoded into the subspace and transmitted through the channel will be high. The monogamy of quantum correlations finally implies that the environment of the channel cannot be significantly coupled to the entanglement which, concluding, ensures the existence of a decoding by the receiver.