Large Deviation Asymptotics Research Articles

Block empirical likelihood inference for stochastic bounding: large deviations asymptotics under m-dependence

Block empirical likelihood inference for stochastic bounding: large deviations asymptotics under m-dependence

A sufficient condition for the quasipotential to be the rate function of the invariant measure of countable-state mean-field interacting particle systems

AbstractThis paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin–Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite-horizon considerations. However, there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin–Wentzell quasipotential is indeed the rate function.

Precise large deviations of aggregate claims in bidimensional risk model with dependence structures

In this article, we consider a bidimensional risk model with strong asymptotic independence between marginal claims having consistent varying tails and study the asymptotics of precise large deviations of aggregate claims by allowing claim-size vectors and their inter-arrival times to be arbitrarily dependent. This study can provide novel insights into the various dependence structures imposed on the involved modeling components.

Achievable Refined Asymptotics for Successive Refinement Using Gaussian Codebooks

We study the mismatched successive refinement problem where one uses Gaussian codebooks to compress an arbitrary memoryless source with successive minimum Euclidean distance encoding under the quadratic distortion measure. Specifically, we derive achievable refined asymptotics under both the joint excess-distortion probability (JEP) and the separate excess-distortion probabilities (SEP) criteria. For both second-order and moderate deviations asymptotics, we consider two types of codebooks: the spherical codebook where each codeword is drawn independently and uniformly from the surface of a sphere and the i.i.d. Gaussian codebook where each component of each codeword is drawn independently from a Gaussian distribution. We establish the achievable second-order rate-region under JEP and we show that under SEP any memoryless source satisfying mild moment conditions is strongly successively refinable. When specialized to a Gaussian memoryless source (GMS), our results provide an alternative achievability proof with specific code design. We show that under JEP and SEP, the same moderate deviations constant is achievable. For large deviations asymptotics, we only consider the i.i.d. Gaussian codebook since the i.i.d. Gaussian codebook has better performance than the spherical codebook in this regime for the one layer mismatched rate-distortion problem (Zhou, Tan, Motani, TIT, 2019). We derive achievable exponents of both JEP and SEP and specialize our results to a GMS, which appears to be a novel result of independent interest.

Large‐scale asymptotics of velocity‐jump processes and nonlocal Hamilton–Jacobi equations

AbstractWe investigate a simple velocity jump process in the regime of large deviation asymptotics. New velocities are taken randomly at a constant, large, rate from a Gaussian distribution with vanishing variance. The Kolmogorov forward equation associated with this process is the linear BGK kinetic transport equation. We derive a new type of Hamilton–Jacobi equation which is nonlocal with respect to the velocity variable. We introduce a suitable notion of viscosity solution, and we prove well‐posedness in the viscosity sense. We also prove convergence of the logarithmic transformation toward this limit problem. Furthermore, we identify the variational formulation of the solution by means of an action functional supported on piecewise linear curves. As an application of this theory, we compute the exact rate of acceleration in a kinetic version of the celebrated F–KPP equation in the one‐dimensional case.

A Kesten–Stigum type theorem for a supercritical multitype branching process in a random environment

Consider a supercritical d-type branching process $Z_n^{i} = $ $ (Z_n^i(1), \cdots, Z_n^i(d)), \,n\geq 0,$ in an i.i.d. environment $\xi =(\xi_0, \xi_1, \ldots)$, starting with one particle of type $i$, whose offspring distributions of generation $n$ depend on the environment $\xi_n$ at time $n$. In the deterministic environment case, the famous Kesten-Stigum (1966) theorem states essentially that, if the mean matrix of the offspring distribution has spectral radius $\rho >1$, then for all $i,j =1, \ldots, d$, almost surely $ \lim_{n\rightarrow \infty} \frac{Z_{n}^{i}(j)}{\rho^n }$ exists and is finite; moreover, the limit variables are non-degenerate if and only if $\mathbb{E} \left(Z_1^{i}(j) \log^{+} Z_1^{i}(j)\right)<+\infty$ for all $i$ and $j$. The extension to the random environment case with $d=1$ has been done by Athreya and Karlin (1971) and Tanny (1988). Extending the Kesten-Stigum theorem to the random environment case with $d>1$ is a long-standing problem. The main objective of this paper is to resolve this delicate problem. In particular, under simple conditions, we prove that for any $1\le i,j\le d$, as $n \rightarrow +\infty$, $Z_n^{i}(j)/\mathbb{E}_\xi Z_n^{i}(j) \rightarrow W^{i}$ in probability, where $W^i$ is a non-negative random variable, $ \mathbb{E}_\xi Z_n^{i}(j)$ is the conditional expectation of $Z_n^{i}(j)$ given the environment $\xi$, which diverges to $\infty$ with geometric rate in the sense that $ \frac{1}{n} \log \mathbb{E}_\xi Z_n^{i}(j) \rightarrow \gamma >0$ almost surely, $\gamma$ being the Lyapunov exponent of the mean matrices of the offspring distributions; moreover $W^i$ are non-degenerate for all $i $ if and only if $\mathbb{E}\left(\frac{Z_1^{i}(j)}{M_0(i,j)}\log^{+}\frac{Z_1^{i}(j)}{M_0(i,j)}\right)<+\infty$ for all $i$ and $j$, where $M_0(i,j)$ is the conditioned mean of the number of children of type $j$ produced by a particle of type $i$ at time $0$, given the environment $\xi$. The key idea of the proof is the introduction of a non-negative martingale $(W_n^{i})$ which converges a.s. to $W^i$, and which reduces to the well-known fundamental martingale in the deterministic environment case. In addition, we prove that the direction $Z_n^{i}/ \| Z_n^{i} \|$ converges in law conditioned on the explosion event $\{\|Z_n^i\| \rightarrow +\infty\}$. The case of stationary and ergodic environment is also considered. Our results open ways to prove important properties such as central limit theorems with convergence rate and large deviation asymptotics.

Return-time -spectrum for equilibrium states with potentials of summable variation

AbstractLet $(X_k)_{k\geq 0}$ be a stationary and ergodic process with joint distribution $\mu $ , where the random variables $X_k$ take values in a finite set $\mathcal {A}$ . Let $R_n$ be the first time this process repeats its first n symbols of output. It is well known that $({1}/{n})\log R_n$ converges almost surely to the entropy of the process. Refined properties of $R_n$ (large deviations, multifractality, etc) are encoded in the return-time $L^q$ -spectrum defined as provided the limit exists. We consider the case where $(X_k)_{k\geq 0}$ is distributed according to the equilibrium state of a potential with summable variation, and we prove that where $P((1-q)\varphi )$ is the topological pressure of $(1-q)\varphi $ , the supremum is taken over all shift-invariant measures, and $q_\varphi ^*$ is the unique solution of $P((1-q)\varphi ) =\sup _\eta \int \varphi \,d\eta $ . Unexpectedly, this spectrum does not coincide with the $L^q$ -spectrum of $\mu _\varphi $ , which is $P((1-q)\varphi )$ , and it does not coincide with the waiting-time $L^q$ -spectrum in general. In fact, the return-time $L^q$ -spectrum coincides with the waiting-time $L^q$ -spectrum if and only if the equilibrium state of $\varphi $ is the measure of maximal entropy. As a by-product, we also improve the large deviation asymptotics of $({1}/{n})\log R_n$ .

Some Large Deviation Asymptotics in Small Noise Filtering Problems

We consider nonlinear filters for diffusion processes when the observation and signal noises are small and of the same order. As the noise intensities approach zero, the nonlinear filter can be approximated by a certain variational problem that is closely related to Mortensen's optimization problem(1968). This approximation result can be made precise through a certain Laplace asymptotic formula. In this work we study probabilities of deviations of true filtering estimates from that obtained by solving the variational problem. Our main result gives a large deviation principle for Laplace functionals whose typical asymptotic behavior is described by Mortensen-type variational problems. Proofs rely on stochastic control representations for positive functionals of Brownian motions and Laplace asymptotics of the Kallianpur-Striebel formula.

Large-deviation asymptotics of condition numbers of random matrices

AbstractLet $\mathbf{X}$ be a $p\times n$ random matrix whose entries are independent and identically distributed real random variables with zero mean and unit variance. We study the limiting behaviors of the 2-normal condition number k(p,n) of $\mathbf{X}$ in terms of large deviations for large n, with p being fixed or $p=p(n)\rightarrow\infty$ with $p(n)=o(n)$ . We propose two main ingredients: (i) to relate the large-deviation probabilities of k(p,n) to those involving n independent and identically distributed random variables, which enables us to consider a quite general distribution of the entries (namely the sub-Gaussian distribution), and (ii) to control, for standard normal entries, the upper tail of k(p,n) using the upper tails of ratios of two independent $\chi^2$ random variables, which enables us to establish an application in statistical inference.

Integro-local limit theorems for supercritical branching process in a random environment

Let Zn be a supercritical branching process in a random environment (BPRE). Under certain moment assumptions, we present the precise asymptotics for the “integro-local” probabilities P(logZn∈[x(n),x(n)+Δn)), where Δn→0 and x(n)→∞ as n→∞. In particular, this implies the large deviations tail asymptotics for P(logZn⩾x(n)) as n→∞. Like in previous research, we can see that, in the light-tail case, the main term in the large deviations asymptotics for the BPRE is provided by the associated random walk.

Precise asymptotics: Robust stochastic volatility models

We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of Bayer et al. (Math. Finance 30 (2020) 782–832) In essence, we implement a Laplace method on the space of models (in the sense of Hairer), which generalizes classical works of Azencott and Ben Arous on path space and then Aida, Inahama–Kawabi on rough path space. When applied to rough volatility models, for example, in the setting of Bayer, Friz and Gatheral (Quant. Finance 16 (2016) 887–904) and Forde–Zhang (SIAM J. Financial Math. 8 (2017) 114–145), one obtains precise asymptotics for European options which refine known large deviation asymptotics.

Fluctuation Theory in the Boltzmann–Grad Limit

We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behaviour are described by dynamical correlations, which can be fully characterized for short times. This provides both a fluctuating Boltzmann equation and large deviation asymptotics.

Precise large deviation asymptotics for products of random matrices

Let (gn)n⩾1 be a sequence of independent identically distributed d×d real random matrices with Lyapunov exponent λ. For any starting point x on the unit sphere in Rd, we deal with the norm |Gnx|, where Gn≔gn…g1. The goal of this paper is to establish precise asymptotics for large deviation probabilities P(log|Gnx|⩾n(q+l)), where q>λ is fixed and l is vanishing as n→∞. We study both invertible matrices and positive matrices and give analogous results for the couple (Xnx,log|Gnx|) with target functions, where Xnx=Gnx∕|Gnx|. As applications we improve previous results on the large deviation principle for the matrix norm ‖Gn‖ and obtain a precise local limit theorem with large deviations.

Goodness-of-Fit Tests Based on Characterization of Uniformity by the Ratio of Order Statistics, and Their Efficiency

We construct integral and supremum type goodness-of-fit tests for the uniform law based on Ahsanullah’s characterization of the uniform law. We discuss limiting distributions of new tests and describe the logarithmic large deviation asymptotics of test statistics under null-hypothesis. This enables us to calculate their local Bahadur efficiency under some parametric alternatives. Conditions of local optimality of new statistics are given.

Large deviations for interacting particle systems: joint mean-field and small-noise limit

We consider a system of stochastic interacting particles in $\mathbb {R}^{d}$ and we describe large deviation asymptotics in a joint mean-field and small-noise limit. Precisely, a large deviation principle (LDP) is established for the empirical measure and the stochastic current, as the number of particles tends to infinity and the noise vanishes, simultaneously. We give a direct proof of the LDP using tilting and subsequently exploiting the link between entropy and large deviations. To this aim we employ consistency of suitable deterministic control problems associated to the stochastic dynamics.

Computation of Extreme Values of Time Averaged Observables in Climate Models with Large Deviation Techniques

One of the goals of climate science is to characterize the statistics of extreme and potentially dangerous events in the present and future climate. Extreme events like heat waves, droughts, or floods due to persisting rains are characterized by large anomalies of the time average of an observable over a long time. The framework of Donsker-Varadhan large deviation theory could therefore be useful for their analysis. In this paper we discuss how concepts and numerical algorithms developed in relation with large deviation theory can be applied to study extreme, rare fluctuations of time averages of surface temperatures at regional scale with comprehensive numerical climate models. We study the convergence of large deviation functions for the time averaged European surface temperature obtained with direct numerical simulation of the climate model Plasim, and discuss their climate implications. We show how using a rare event algorithm can improve the efficiency of the computation of the large deviation rate functions. We discuss the relevance of the large deviation asymptotics for applications, and we show how rare event algorithms can be used also to improve the statistics of events on time scales shorter than the one needed for reaching the large deviation asymptotics.

Non-Asymptotic Converse Bounds and Refined Asymptotics for Two Source Coding Problems

In this paper, we revisit two multi-terminal lossy source coding problems: the lossy source coding problem with side information available at the encoder and one of the two decoders, which we term as the Kaspi problem (Kaspi, 1994), and the multiple description coding problem with one semi-deterministic distortion measure, which we refer to as the Fu-Yeung problem (Fu and Yeung, 2002). For the Kaspi problem, we first present the properties of optimal test channels. Subsequently, we generalize the notion of the distortion-tilted information density for the lossy source coding problem to the Kaspi problem and prove a non-asymptotic converse bound using the properties of optimal test channels and the well-defined distortion-tilted information density. Finally, for discrete memoryless sources, we derive refined asymptotics which includes the second-order, large, and moderate deviations asymptotics. In the converse proof of second-order asymptotics, we apply the Berry-Esseen theorem to the derived non-asymptotic converse bound. The achievability proof follows by first proving a type-covering lemma tailored to the Kaspi problem, then properly Taylor expanding the well-defined distortion-tilted information densities and finally applying the Berry-Esseen theorem. We then generalize the methods used in the Kaspi problem to the Fu-Yeung problem. As a result, we obtain the properties of optimal test channels for the minimum sum-rate function, a non-asymptotic converse bound and refined asymptotics for discrete memoryless sources. Since the successive refinement problem is a special case of the Fu-Yeung problem, as a by-product, we obtain a non-asymptotic converse bound for the successive refinement problem, which is a strict generalization of the non-asymptotic converse bound for successively refinable sources (Zhou, Tan, and Motani, 2017).

Refined Asymptotics for Rate-Distortion Using Gaussian Codebooks for Arbitrary Sources

The rate-distortion saddle-point problem considered by Lapidoth (1997) consists in finding the minimum rate to compress an arbitrary ergodic source when one is constrained to use a random Gaussian codebook and minimum (Euclidean) distance encoding is employed. We extend Lapidoth’s analysis in several directions in this paper. First, we consider refined asymptotics. In particular, when the source is stationary and memoryless, we establish the second-order, moderate, and large deviation asymptotics of the problem. Second, by random Gaussian codebook, Lapidoth referred to a collection of random codewords, each of which is drawn independently and uniformly from the surface of an $n$ -dimensional sphere. To be more precise, we term this as a spherical codebook. We also consider i.i.d. Gaussian codebooks in which each random codeword is drawn independently from a product Gaussian distribution. We derive the second-order, moderate, and large deviation asymptotics when i.i.d. Gaussian codebooks are employed. In contrast to the recent work on the channel coding counterpart by Scarlett, Tan, and Durisi (2017), the dispersions for spherical and i.i.d. Gaussian codebooks are identical. The ensemble excess-distortion exponents for both spherical and i.i.d. Gaussian codebooks are established for all rates. Furthermore, we show that the i.i.d. Gaussian codebook has a strictly larger excess-distortion exponent than its spherical counterpart for any rate greater than the ensemble rate-distortion function derived by Lapidoth.

On Large Deviations of Interface Motions for Statistical Mechanics Models

We discuss the sharp interface limit of the action functional associated to either the Glauber dynamics for Ising systems with Kac potentials or the Glauber+Kawasaki process. The corresponding limiting functionals, for which we provide explicit formulae of the mobility and transport coefficients, describe the large deviations asymptotics with respect to the mean curvature flow.

Exact moderate and large deviations for linear random fields

Abstract By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.