Large Deviation Rate Research Articles

Landscapes of non-gradient dynamics without detailed balance: Stable limit cycles and multiple attractors

Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.

Variational problem in the non-negative orthant of ℝ3: reflective faces and boundary influence cones

In this paper we consider the variational problem in the non-negative orthant of ?3. The solution of this problem gives the large deviation rate function for the stationary distribution of an SRBM (Semimartingal Reflecting Brownian Motion). Avram, Dai and Hasenbein (Queueing Syst. 37, 259---289, 2001) provided an explicit solution of this problem in the non-negative quadrant. Building on this work, we characterize reflective faces of the non-negative orthant of ? d , we construct boundary influence cones and we provide an explicit solution of several constrained variational problems in ?3. Moreover, we give conditions under which certain spiraling paths to a point on an axis have a cost which is strictly less than the cost of every direct path and path with two pieces.

A representation formula for large deviations rate functionals of invariant measures on the one dimensional torus

Dans cet article, nous considerons une diffusion sur le tore de dimension 1 pour laquelle nous etablissons une formule simple pour la fonction de taux d’un principe de grandes deviations pour sa mesure invariante, lorsque le bruit tend vers 0. Cette fonction de taux a ete precedemment caracterisee par M. I. Freidlin et A. D. Wentzell comme la solution d’un probleme d’optimisation plutot complique. En reduisant ce probleme, nous etablissons notre formule a travers une transformation geometrique qui, a l’aide d’une construction de type Maxwell, cree des regions plates. Nous considerons egalement des processus de Markov deterministes par morceaux, sur le tore de dimension 1. Nous montrons que la fonction de taux, dans un principe de grandes deviations pour la mesure invariante, peut etre obtenue en considerant de nouveau la meme transformation geometrique. Inspires par ces similarites, nous prouvons un resultat d’universalite de cette transformation en montrant qu’elle engendre la solution de viscosite d’une equation d’Hamilton–Jacobi stationnaire, associee a un Hamiltonien H qui satisfait a certaines hypotheses minimales.

Chebyshev-Type Exponential Inequalities for Sums of Random Vectors and for Trajectories of Random Walks

We obtain analogues of the well-known Chebyshev's exponential inequality ${\bf P}(\xi \ge x)\le e^{-\Lambda^{(\xi)}(x)}$, $x>{\bf E}\,\xi,$ for the distribution of a random variable $\xi$, where $\Lambda^{(\xi)}(x):=\sup_\lambda\{\lambda x- \log {\bf E}\,e^{\lambda \xi}\}$ is the large deviation rate function for $\xi$. Generalizations of this relation are established for multivariate random vectors $\xi$, for sums of the vectors, and for trajectories of random processes associated with such sums.

Large deviations and slowdown asymptotics for one-dimensional excited random walks

We study the large deviations of excited random walks on $\mathbb{Z}$. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed $v_0$, then the large deviation rate function for the position of the excited random walk is zero on the interval $[0,v_0]$ and so probabilities such as $P(X_n 2$ is the expected total drift per site of the cookie environment.

Large Deviation Results for Wave Governed Random Motions Driven by Semi-Markov Processes

In this article, we present large deviation results for a model {ξ1 + … + ξ n : n ≥ 1} which is close to a random walk. More precisely, we consider independent random variables {ξ n : n ≥ 1} such that {ξ n : n ≥ 2} are i.i.d. and a different distribution for ξ1 is allowed. We prove large deviation estimates for P(N x ≤ xT) and P(N x < ∞) as x → ∞, where N x : = inf {n ≥ 1: ξ1 + … + ξ n ≥ x}. Moreover, we provide an asymptotically efficient simulation law for the estimation of P(N x ≤ xT) and P(N x < ∞) by Monte Carlo simulation based on the importance sampling technique. These results will be adapted to wave governed random motions driven by semi-Markov processes and we present some simulations. Finally, we study the convergence of some large deviation rates for standard wave governed random motions based on a scaling presented in the literature (see Kac, 1974; Orsingher, 1990).

From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage

We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h > 0, a large-deviations rate functional J h characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the time-discretized entropy-Wasserstein gradient flow is characterized by the minimization of a functional K h . We establish a new connection between these systems by proving that J h and K h are equal up to second order in h as h → 0. This result gives a microscopic explanation of the origin of the entropy-Wasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.

Asymptotic analysis of the optimal cost in some transportation problems with random locations

In this paper we provide an asymptotic analysis of the optimal transport cost in some matching problems with random locations. More precisely, under various assumptions on the distribution of the locations and the cost function, we prove almost sure convergence, and large and moderate deviation principles. In general, the rate functions are given in terms of infinite-dimensional variational problems. For a suitable one-dimensional transportation problem, we provide the expression of the large deviation rate function in terms of a one-dimensional optimization problem, which allows the numerical estimation of the rate function. Finally, for certain one-dimensional transportation problems, we prove a central limit theorem.

Large deviations for renewal processes

We investigate large deviations for the empirical measure of the forward and backward recurrence time processes associated with a classical renewal process with arbitrary waiting-time distribution. The Donsker–Varadhan theory cannot be applied in this case, and indeed it turns out that the large deviations rate functional differs from the one suggested by such a theory. In particular, a non-strictly convex and non-analytic rate functional is obtained.

How many eigenvalues of a Gaussian random matrix are positive?

We study the probability distribution of the index N(+), i.e., the number of positive eigenvalues of an N×N Gaussian random matrix. We show analytically that, for large N and large N(+) with the fraction 0≤c=N(+)/N≤1 of positive eigenvalues fixed, the index distribution P(N(+)=cN,N)~exp[-βN(2)Φ(c)] where β is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function Φ(c) is computed explicitly for all 0≤c≤1. It is independent of β and displays a quadratic form modulated by a logarithmic singularity around c=1/2. As a consequence, the distribution of the index has a Gaussian form near the peak, but with a variance Δ(N) of index fluctuations growing as Δ(N)~lnN/βπ(2) for large N. For β=2, this result is independently confirmed against an exact finite-N formula, yielding Δ(N)=lnN/2π(2)+C+O(N(-1)) for large N, where the constant C for even N has the nontrivial value C=(γ+1+3ln2)/2π(2)≃0.185 248… and γ=0.5772… is the Euler constant. We also determine for large N the probability that the interval [ζ(1),ζ(2)] is free of eigenvalues. Some of these results have been announced in a recent letter [Phys. Rev. Lett. 103, 220603 (2009)].

On the large deviations of a class of modulated additive processes

We prove that the large deviation principle holds for a class of processes inspired by semi-Markov additive processes. For the processes we consider, the sojourn times in the phase process need not be independent and identically distributed. Moreover the state selection process need not be independent of the sojourn times. We assume that the phase process takes values in a finite set and that the order in which elements in the set, called states, are visited is selected stochastically. The sojourn times determine how long the phase process spends in a state once it has been selected. The main tool is a representation formula for the sample paths of the empirical laws of the phase process. Then, based on assumed joint large deviation behavior of the state selection and sojourn processes, we prove that the empirical laws of the phase process satisfy a sample path large deviation principle. From this large deviation principle, the large deviations behavior of a class of modulated additive processes is deduced. As an illustration of the utility of the general results, we provide an alternate proof of results for modulated Levy processes. As a practical application of the results, we calculate the large deviation rate function for a processes that arises as the International Telecommunications Union's standardized stochastic model of two-way conversational speech.

Differing Averaged and Quenched Large Deviations for Random Walks in Random Environments in Dimensions Two and Three

We consider the quenched and the averaged (or annealed) large deviation rate functions $I_q$ and $I_a$ for space-time and (the usual) space-only RWRE on $\mathbb{Z}^d$. By Jensen's inequality, $I_a\leq I_q$. In the space-time case, when $d\geq3+1$, $I_q$ and $I_a$ are known to be equal on an open set containing the typical velocity $\xi_o$. When $d=1+1$, we prove that $I_q$ and $I_a$ are equal only at $\xi_o$. Similarly, when d=2+1, we show that $I_a<I_q$ on a punctured neighborhood of $\xi_o$. In the space-only case, we provide a class of non-nestling walks on $\mathbb{Z}^d$ with d=2 or 3, and prove that $I_q$ and $I_a$ are not identically equal on any open set containing $\xi_o$ whenever the walk is in that class. This is very different from the known results for non-nestling walks on $\mathbb{Z}^d$ with $d\geq4$.

Novel bandwidth requirement estimation based on exact large deviation asymptotics

This paper deals with bandwidth requirement estimators in buffered statistical multiplexers with the underlying large deviation many sources asymptotics modeling framework. Within this multiplexer model asymptotically exact approximation for buffered workload loss ratio is considered. Based on this, a direct and computationally feasible equivalent capacity estimate is presented in full accordance with the corresponding QoS constraints. This means that an equation involving a function of the large deviation rate function is solved symbolically with respect to the bandwidth requirement estimate. Analytical considerations and numerical investigations illustrate the usefulness of the novel estimate.

Bose–Einstein condensation in large deviations with applications to informationsystems

We study the large deviation behavior of systems that admit a certain form of a productdistribution, which is frequently encountered both in physics and in various informationsystem models. First, to fix ideas, we demonstrate a simple calculation of the largedeviations rate function for a single constraint (event). Under certain conditions, thebehavior of this function is shown to exhibit an analog of Bose–Einstein condensation(BEC). More interestingly, we also study the large deviations rate function associated withtwo constraints (and the extension to any number of constraints is conceptuallystraightforward). The phase diagram of this rate function is shown to exhibit as many asseven phases, and it suggests a two-dimensional generalization of the notion of BEC (ormore generally, a multi-dimensional BEC). While the results are illustrated fora simple model, the underlying principles are actually rather general. We alsodiscuss several applications and implications pertaining to information systemmodels.

Large deviations for stochastic flows of diffeomorphisms

A large deviation principle is established for a general class of stochastic flows in the small noise limit. This result is then applied to a Bayesian formulation of an image matching problem, and an approximate maximum likelihood property is shown for the solution of an optimization problem involving the large deviations rate function.

Censored Exponential Data: Large Deviations for MLEs and Posterior Distributions

In this article, we consider several statistical models for censored exponential data. We prove a large deviation result for the maximum likelihood estimators (MLEs) of each model, and a unique result for the posterior distributions which works well for all the cases. Finally, comparing the large deviation rate functions for MLEs and posterior distributions, we show that a typical feature fails for one model; moreover, we illustrate the relation between this fact and a well-known result for curved exponential models.

Law of Large Numbers for Wiener Measure with Density Having Two Large Deviation Minimizers

This paper discusses the situation that the large deviation rate functional has two distinct minimizers, for a model described by Wiener measures with certain densities involving a scaling. The motivation comes from the study of the so-called $\nabla\varphi$ interface model with weak self potentials. The pinned Wiener measures case was discussed by [3].

Importance sampling for Jackson networks

Rare event simulation in the context of queueing networks has been an active area of research for more than two decades. A commonly used technique to increase the efficiency of Monte Carlo simulation is importance sampling. However, there are few rigorous results on the design of efficient or asymptotically optimal importance sampling schemes for queueing networks. Using a recently developed game/subsolution approach, we construct simple and efficient state-dependent importance sampling schemes for simulating buffer overflows in stable open Jackson networks. The sampling distributions do not depend on the particular event of interest, and hence overflow probabilities for different events can be estimated simultaneously. A by-product of the analysis is the identification of the minimizing trajectory for the calculus of variation problem that is associated with the sample-path large deviation rate function.

Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution

Let Z be a two-dimensional Brownian motion confined to the non-negative quadrant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for two-station queueing networks. The parameters of Z are a drift vector, a covariance matrix, and a "direction of reflection" for each of the quadrant's two boundary rays. Necessary and sufficient conditions are known for Z to be a positive recurrent semimartingale, and they are the only restrictions imposed on the process data in our study. Under those assumptions, a large deviations principle (LDP) is conjectured for the stationary distribution of Z, and we recapitulate the cases for which it has been rigorously justified. For sufficiently regular sets B, the LDP says that the stationary probability of xB decays exponentially as x??, and the asymptotic decay rate is the minimum value achieved by a certain function I(?) over the set B. Avram, Dai and Hasenbein (Queueing Syst.: Theory Appl. 37, 259---289, 2001) provided a complete and explicit solution for the large deviations rate function I(?). In this paper we re-express their solution in a simplified form, showing along the way that the computation of I(?) reduces to a relatively simple problem of least-cost travel between a point and a line.

Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems

We study sample covariance matrices of the form W = (1 / n)CCT, where C is a k x n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-called Wishart matrices, where the entries of C are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞ or kn → ∞ with kn = o(n / log log n), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving almost sure limits of the eigenvalues, require only finite fourth moments. Our most explicit results for large k are for the case where the entries of C are ∓ 1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalues to the rate functions for i.i.d. standard normal entries of C. This case is of particular interest since it is related to the problem of decoding of a signal in a code-division multiple-access (CDMA) system arising in mobile communication systems. In this example, k is the number of users in the system and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency; the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.