Large Deviation Result Research Articles

Multidimensional strong large deviation results

We establish strong large deviation results for an arbitrary sequence of random vectors under some assumptions on the normalized cumulant generating function. In other words, we give asymptotic approximations for a multivariate tail probability of the same kind as the one obtained by Bahadur and Rao (Ann Math Stat 31:1015–1027, 1960) for the sample mean (in the one-dimensional case). The proof of our results follows the same lines as in Chaganty and Sethuraman (J Stat Plan Inference, 55:265–280, 1996). We also present three statistical applications to illustrate our results, the first one dealing with a vector of independent sample variances, the second one with a Gaussian multiple linear regression model and the third one with the multivariate Nadaraya–Watson estimator. Some numerical results are also presented for the first two applications.

On the distribution of the van der Corput sequence in arbitrary base

A central limit theorem with explicit error bound, and a large deviation result are proved for a sequence of weakly dependent random variables of a special form. As a corollary, under certain conditions on the function \(f:[0,1] \rightarrow \mathbb {R}\) a central limit theorem and a large deviation result are obtained for the sum \(\sum _{n=0}^{N-1} f(x_n)\), where \(x_n\) is the base b van der Corput sequence for an arbitrary integer \(b \ge 2\). Similar results are also proved for the \(L^p\) discrepancy of the same sequence for \(1 \le p < \infty \). The main methods used in the proofs are the Berry–Esseen theorem and Fourier analysis.

Fluctuation theorems for discrete kinetic models of molecular motors

Motivated by discrete kinetic models for non-cooperative molecular motors on periodic tracks, we consider random walks (also not Markov) on quasi one dimensional (1d) lattices, obtained by gluing several copies of a fundamental graph in a linear fashion. We show that, for a suitable class of quasi-1d lattices, the large deviation rate function associated to the position of the walker satisfies a Gallavotti–Cohen symmetry for any choice of the dynamical parameters defining the stochastic walk. This class includes the linear model considered in Lacoste et al (2008 Phys. Rev. E 78 011915). We also derive fluctuation theorems for the time-integrated cycle currents and discuss how the matrix approach of Lacoste et al (2008 Phys. Rev. E 78 011915) can be extended to derive the above Gallavotti–Cohen symmetry for any Markov random walk on with periodic jump rates. Finally, we review in the present context some large deviation results of Faggionato and Silvestri (2017 Ann. Inst. Henri Poincaré 53 46–78) and give some specific examples with explicit computations.

Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables.

Recent large deviation results have provided general lower bounds for the fluctuations of time-integrated currents in the steady state of stochastic systems. A corollary are so-called thermodynamic uncertainty relations connecting precision of estimation to average dissipation. Here we consider this problem but for counting observables, i.e., trajectory observables which, in contrast to currents, are non-negative and nondecreasing in time (and possibly symmetric under time reversal). In the steady state, their fluctuations to all orders are bound from below by a Conway-Maxwell-Poisson distribution dependent only on the averages of the observable and of the dynamical activity. We show how to obtain the corresponding bounds for first-passage times (times when a certain value of the counting variable is first reached) and their uncertainty relations. Just like entropy production does for currents, dynamical activity controls the bounds on fluctuations of counting observables.

Large deviations for estimators of the parameters of a neuronal response latency model

We consider a model in the literature for the neuronal activity with response latency. We present large deviation results for two sequences of estimators of some unknown parameters. We also present a large deviation result for the posterior distributions in the Bayesian setting.

Harmonic moments and large deviations for a supercritical branching process in a random environment

Let $(Z_n)_{n\geq 0}$ be a supercritical branching process in an independent and identically distributed random environment $\xi =(\xi _n)_{n\geq 0}$. We study the asymptotic behavior of the harmonic moments $\mathbb{E} \left [Z_n^{-r} | Z_0=k \right ]$ of order $r>0$ as $n \to \infty $, when the process starts with $k$ initial individuals. We exhibit a phase transition with the critical value $r_k>0$ determined by the equation $\mathbb E p_1^k(\xi _0) = \mathbb E m_0^{-r_k},$ where $m_0=\sum _{j=0}^\infty j p_j (\xi _0)$, $(p_j(\xi _0))_{j\geq 0}$ being the offspring distribution given the environnement $\xi _0$. Contrary to the constant environment case (the Galton-Watson case), this critical value is different from that for the existence of the harmonic moments of $W=\lim _{n\to \infty } Z_n / \mathbb E (Z_n|\xi ).$ The aforementioned phase transition is linked to that for the rate function of the lower large deviation for $Z_n$. As an application, we obtain a lower large deviation result for $Z_n$ under weaker conditions than in previous works and give a new expression of the rate function. We also improve an earlier result about the convergence rate in the central limit theorem for $W-W_n,$ and find an equivalence for the large deviation probabilities of the ratio $Z_{n+1} / Z_n$.

Kardar–Parisi–Zhang Equation and Large Deviations for Random Walks in Weak Random Environments

We consider the transition probabilities for random walks in $1+1$ dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.

Large deviations for the Ornstein–Uhlenbeck process without tears

Our goal is to establish large deviations for the maximum likelihood estimator of the drift parameter of the Ornstein–Uhlenbeck process without tears. We propose a new strategy to establish large deviation results which allows us, via a suitable transformation, to circumvent the classical difficulty of non-steepness. Our approach holds in the stable case where the process is positive recurrent as well as in the unstable and explosive cases where the process is respectively null recurrent and transient. It can also be successfully implemented for more complex diffusion processes.

What to Expect When You Are Expecting on the Grassmannian

Consider an incoming sequence of vectors, all belonging to an unknown subspace $\operatorname{S}$, and each with many missing entries. In order to estimate $\operatorname{S}$, it is common to partition the data into blocks and iteratively update the estimate of $\operatorname{S}$ with each new incoming measurement block. In this paper, we investigate a rather basic question: Is it possible to identify $\operatorname{S}$ by averaging the column span of the partially observed incoming measurement blocks on the Grassmannian? We show that in general the span of the incoming blocks is in fact a biased estimator of $\operatorname{S}$ when data suffers from erasures, and we find an upper bound for this bias. We reach this conclusion by examining the defining optimization program for the Fr\'{e}chet expectation on the Grassmannian, and with the aid of a sharp perturbation bound and standard large deviation results.

Rough differential equations with unbounded drift term

We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly. Furthermore, we show that the semiflow exists assuming only appropriate one-sided growth conditions. We provide bounds for both the flow and the semiflow. Applied to stochastic analysis, our results imply strong completeness and the existence of a stochastic (semi)flow for a large class of stochastic differential equations. If the driving process is Gaussian, we can further deduce (essentially) sharp tail estimates for the (semi)flow and a Freidlin–Wentzell-type large deviation result.

Dispersion in Rectangular Networks: Effective Diffusivity and Large-Deviation Rate Function.

The dispersion of a diffusive scalar in a fluid flowing through a network has many applications including to biological flows, porous media, water supply, and urban pollution. Motivated by this, we develop a large-deviation theory that predicts the evolution of the concentration of a scalar released in a rectangular network in the limit of large time t≫1. This theory provides an approximation for the concentration that remains valid for large distances from the center of mass, specifically for distances up to O(t) and thus much beyond the O(t^{1/2}) range where a standard Gaussian approximation holds. A byproduct of the approach is a closed-form expression for the effective diffusivity tensor that governs this Gaussian approximation. MonteCarlo simulations of Brownian particles confirm the large-deviation results and demonstrate their effectiveness in describing the scalar distribution when t is only moderately large.

Approximation by Normal Distribution for a Sample Sum in Sampling Without Replacement from a Finite Population

A sum of observations derived by a simple random sampling design from a population of independent random variables is studied. A procedure finding a general term of Edgeworth asymptotic expansion is presented. The Lindeberg condition of asymptotic normality, Berry-Esseen bound, Edgeworth asymptotic expansions under weakened conditions and Cramer type large deviation results are derived.

Joint large deviation result for empirical measures of the coloured random geometric graphs.

We prove joint large deviation principle for the empirical pair measure and empirical locality measure of the near intermediate coloured random geometric graph models on n points picked uniformly in a d-dimensional torus of a unit circumference. From this result we obtain large deviation principles for the number of edges per vertex, the degree distribution and the proportion of isolated vertices for the near intermediate random geometric graph models.

The inner boundary of random walk range

In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If $L_n$ be the number of the inner boundary points of random walk range in the $n$ steps, we prove $\lim_{n\to \infty} ({L_n}/{n})$ exists with probability one. Also, we obtain some large deviation result for transient walk. We find that the expectation of the number of the inner boundary points of simple random walk on the two dimensional square lattice is of the same order as ${n}/{(\log n)^2}$.

An introduction to large deviations for random graphs

This article gives an overview of the emerging literature on large deviations for random graphs. Written for the general mathematical audience, the article begins with a short introduction to the theory of large deviations. This is followed by a description of some large deviation questions about random graphs, and an outline of the recent progress on this topic. A more elaborate discussion follows, with a brief account of graph limit theory and its application in constructing a large deviation theory for dense random graphs. The role of Szemer\'edi's regularity lemma is explained, together with a sketch of the proof of the main large deviation result and some examples. Applications to exponential random graph models are briefly touched upon. The remainder of the paper is devoted to large deviations for sparse graphs. Since the regularity lemma is not applicable in the sparse regime, new tools are needed. Fortunately, there have been several new breakthroughs that managed to achieve the goal by an indirect method. These are discussed, together with an exposition of the underlying theory. The last section contains a list of open problems.

The large deviation for the least squares estimator of nonlinear regression model based on WOD errors

As a kind of dependent random variables, the widely orthant dependent random variables, or WOD for short, have a very important place in dependence structures for the intricate properties. And so its behavior and properties in different statistical models will be a major part in our research interest. Based on WOD errors, the large deviation results of the least squares estimator in the nonlinear regression model are established, which extend the corresponding ones for independent errors and some dependent errors.

Asymptotic results for multivariate estimators of the mean density of random closed sets

The problem of the evaluation and estimation of the mean density of random closed sets in Rd with integer Hausdorff dimension 0 < n < d, is of great interest in many different scientific and technological fields. Among the estimators of the mean density available in literature, the so-called “Minkowski content”-based estimator reveals its benefits in applications in the non-stationary cases. We introduce here a multivariate version of such estimator, and we study its asymptotical properties by means of large and moderate deviation results. In particular we prove that the estimator is strongly consistent and asymptotically Normal. Furthermore we also provide confidence regions for the mean density of the involved random closed set in m ≥ 1 distinct points x1, . . . , xm ∈ Rd.

Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation

This is an expository paper that reviews recent developments on optimal estimation of structured high-dimensional covariance and precision matrices. Minimax rates of convergence for estimating several classes of structured covariance and precision matrices, including bandable, Toeplitz, and sparse covariance matrices as well as sparse precision matrices, are given under the spectral norm loss. Data-driven adaptive procedures for estimating various classes of matrices are presented. Some key technical tools including large deviation results and minimax lower bound arguments that are used in the theoretical analyses are discussed. In addition, estimation under other losses and a few related problems such as Gaussian graphical models, sparse principal component analysis, and hypothesis testing on the covariance structure are considered. Some open problems on estimating high-dimensional covariance and precision matrices and their functionals are also discussed.

The dynamics of the stochastic shadow Gierer–Meinhardt system

We consider the dynamics of the stochastic shadow Gierer–Meinhardt system with one-dimensional standard Brownian motion. We establish the global existence and uniqueness of solutions. We also prove a large deviation result.

Moderate Deviations for Recursive Stochastic Algorithms

We prove a moderate deviation principle for the continuous time interpolation of discrete time recursive stochastic processes. The methods of proof are somewhat different from the corresponding large deviation result, and in particular the proof of the upper bound is more complicated.