Local Large Deviation Research Articles

Local large deviations for periodic infinite horizon Lorentz gases

We prove optimal local large deviations for the periodic infinite horizon Lorentz gas viewed as a ℤd-cover (d = 1,2) of a dispersing billiard. In addition to this specific example, we prove a general result for a class of nonuni-formly hyperbolic dynamical systems and observables associated with central limit theorems with nonstandard normalisation.

Analytic proof of multivariate stable local large deviations and application to deterministic dynamical systems

We give a short analytic proof of local large deviations for i.i.d. random variables in the domain of a multivariate α-stable law, α∈(0,1)∪(1,2]. Our method simultaneously covers lattice and nonlattice distributions (and mixtures thereof), bypassing aperiodicity considerations. The proof applies also to the dynamical setting.

Large deviations and information theory for sub-critical signal-to-interference-plus-noise ratio random network models

The article obtains large deviation asymptotic for sub-critical communication networks modelled as signal-interference-noise-ratio(SINR) random networks. To achieve this, we define the empirical power measure and the empirical connectivity measure, as well as prove joint large deviation principles(LDPs) for the two empirical measures on two different scales. Using the joint LDPs, we prove an asymptotic equipartition property(AEP) for wireless telecommunication networks modelled as the subcritical SINR random networks. Further, we prove a local large deviation principle(LLDP) for the sub-critical SINR random network. From the LLDPs, we prove the large deviation principle, and a classical McMillan Theorem for the stochastic SINR model processes. Note that, the LDPs for the empirical measures of this stochastic SINR random network model were derived on spaces of measures equipped with the τ- topology, and the LLDPs were deduced in the space of SINR model process without any topological limitations. We motivate the study by describing a possible anomaly detection test for SINR random networks.

Large deviations, asymptotic equipartition property for super-critical SINR random networks

In this article, we obtain large deviation asymptotic for supercritical social or communication networks modelled as signal-interference-noise ratio (SINR) graphs. To do this, we define the empirical power measure and the empirical connectivity measure, and prove joint large deviation principles (LDPs) for the two empirical measures on two different scales, i.e. λ and λ 2 aλ , where λ is the intensity measure of the Poisson point process (PPP) which defines the SINR random network and aλ real-valued sequence such that λ 2 aλ → ∞ as λ → ∞Using these joint LDPs we prove an asymptotic equipartition property for the super-critical networks modelled as the SINR random networks. Furthermore, we prove a local large deviation principle (LLDP) for the SINR random network. From the LLDP we prove a large deviation principle, and a classical McMillian theorem for the SINR random network processes. Note that, for a given empirical power measure and typical empirical connectivity measure, qπ ⊗ π, we can deduce from the LLDP a bound on the cardinality of the space of SINR networks to be approximately equal to , where the connectivity probability of the network, , satisfies and qπ ⊗ π is the typical behavior of the empirical connectivity measure. Observe, the LDPs for the empirical measures of SINR random networks were obtained on spaces of measures equipped with the τ-topology, and the LLDPs were obtained in the space of SINR network process without any topological restrictions.

The local principle of large deviations for compound Poisson process with catastrophes

The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle.

Local Moderate and Precise Large Deviations via Cluster Expansions

We consider a system of classical particles confined in a box \(\varLambda \subset {\mathbb {R}}^d\) with zero boundary conditions interacting via a stable and regular pair potential. Based on the validity of the cluster expansion for the canonical partition function in the high temperature–low density regime we prove moderate and precise large deviations from the mean value of the number of particles with respect to the grand-canonical Gibbs measure. In this way we have a direct method of computing both the exponential rate as well as the pre-factor and obtain explicit error terms. Estimates comparing with the infinite volume versions of the above are also provided.

Local large deviation principle, large deviation principle and information theory for the signal-to-interference-plus-noise ratio graph models

Given devices space D, an intensity measure λm ϵ (0, ∞), a transition kernel Q from the space D to positive real numbers ℝ+, a path-loss function (which depends on the Euclidean distance between the devices and a positive constant a), we define a Marked Poisson Point process (MPPP). For a given MPPP and technical constants , we define a Marked Signal-to- Interference and Noise Ratio (SINR) graph, and associate with it two empirical measures; the empirical marked measure and the empirical connectivity measure. For a class of marked SINR graphs, we prove a joint large deviation principle(LDP) for these empirical measures, with speed λ in the τ-topology. From the joint large deviation principle for the empirical marked measure and the empirical connectivity measure, we obtain an Asymptotic Equipartition Property(AEP) for network structured data modelled as a marked SINR graph. Specifically, we show that for large dense marked SINR graph one requires approximately about λ 2 H(Q × Q)/log 2 bits to transmit the information contained in the network with high probability, where H(Q × Q) is a properly defined entropy for the exponential transition kernel with parameter c. Further, we prove a local large deviation principle (LLDP) for the class of marked SINR graphs on D, where , with speed λ from a spectral potential point. From the LLDP we derive a conditional LDP for the marked SINR graphs. Note that, while the joint LDP is established in the τ-topology, the LLDP assume no topological restriction on the space of marked SINR graphs. Observe also that all our rate functions are expressed in terms of the relative entropy or the kullback action or divergence function of the marked SINR on the devices space D.

A remark on normalizations in a local large deviations principle for inhomogeneous birth-and-death process

A remark on normalizations in a local large deviations principle for inhomogeneous birth-and-death process

A large deviation result for maximum likelihood estimator of non-homogeneous Ornstein–Uhlenbeck processes

We establish the large deviation principle for maximum likelihood estimator of some diffusion process. We overcome the difficulty of non-steepness and obtain large deviations in the case of non-Gaussian limit distribution by local large deviation principle and exponential tightness.

Local large deviation principle for Wiener process with random resetting

We consider a class of Markov processes with resettings, where at random times, the Markov processes are restarted from a predetermined point or a region. These processes are frequently applied in physics, chemistry, biology, economics, and in population dynamics. In this paper, we establish the local large deviation principle (LLDP) for the Wiener processes with random resettings, where the resettings occur at the arrival time of a Poisson process. Here, at each resetting time, a new resetting point is selected at random, according to a conditional distribution.

Sparse-Lagrangian MMC modelling of the Sandia DME flame series

A series of turbulent, piloted dimethyl ether (DME)/air jet flames (Sandia DME flames D–G’), with Reynolds numbers ranging from 29,300 to 73,250, has been simulated using a sparse-Lagrangian multiple mapping conditioning (MMC) approach coupled to a large eddy simulation (LES) flow field solver. Mixing between the Monte-Carlo particles is modelled by a generalised form of MMC combined with a sparse distribution of particles leading to significant computational savings compared to what is required for conventional mixing models. This is achieved by pairwise mixing of particles that are selected dependent on their distance in an extended space comprised of a reference variable, given by the LES mixture fraction, and spatial location. The MMC-LES method successfully predicts the flame structure and composition field for the full flame series. Numerical results are compared against conditional statistics and spatially resolved experimental data acquired with Raman/Rayleigh scattering and laser-induced fluorescence measurements. They show good agreement even for flame DME-G’ where large turbulence-chemistry interactions lead to significant local extinction and large deviations from a flamelet structure. The influence of the mixing time on the predicted flame structure is investigated, and the systematic validation of the time scale models with the aid of measurements of the entire flame series has corroborated the findings of earlier DNS and single flame studies: a modified time scale model is needed to provide accurate predictions of conditional fluctuations and thus of possible deviations from a flamelet-like combustion regime.

Local large deviations and the strong renewal theorem

We establish two different, but related results for random walks in the domain of attraction of a stable law of index $\alpha$. The first result is a local large deviation upper bound, valid for $\alpha \in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $\alpha \in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long standing problem, which dates back to the 1962 paper of Garsia and Lamperti [Comm. Math. Helv.] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Pacific J. Math.] for general random walks. This paper supersedes the individual preprints arXiv:1507.07502 and arXiv:1507.06790

Strong renewal theorems and local large deviations for multivariate random walks and renewals

We study a random walk $\mathbf{S} _n$ on $\mathbb{Z} ^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbol{\alpha } =(\alpha _1,\ldots ,\alpha _d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, i.e. a sharp asymptotic of the Green function $G(\mathbf{0} ,\mathbf{x} )$ as $\|\mathbf{x} \|\to +\infty $, along the “favorite direction or scaling”: (i) if $\sum _{i=1}^d \alpha _i^{-1} < 2$ (reminiscent of Garsia-Lamperti’s condition when $d=1$ [17]); (ii) if a certain local condition holds (reminiscent of Doney’s [13, Eq. (1.9)] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0} ,\mathbf{x} )$, sharpening estimates when $\mathbf{x} $ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\alpha _i\equiv \alpha $, in the favorite scaling, and has even left aside the case $\alpha \in [1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.

Notes on random walks in the Cauchy domain of attraction

The goal of these notes is to fill some gaps in the literature about random walks in the Cauchy domain of attraction, which has been in many cases left aside because of its additional technical difficulties. We prove here several results in that case: a Fuk-Nagaev inequality and a local version of it ; a large deviation theorem ; two types of local large deviation theorems. We also derive two important applications of these results: a sharp estimate of the tail of the first ladder epochs, and renewal theorems -- extending standard renewal theorems to the case of random walks. Most of our techniques carry through to the case of random walks in the domain of attraction of an $\alpha$-stable law with $\alpha \in(0,2)$, so we also present results in that case, since many of them seem to be missing in the literature.

Local large deviations principle for occupation measures of the stochastic damped nonlinear wave equation

We consider the damped nonlinear wave (NLW) equation driven by a noise which is white in time and colored in space. Assuming that the noise is non-degenerate in all Fourier modes, we establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation and is a novelty in the context of randomly forced PDE’s. The proof is based on an extension of methods developed in (Comm. Pure Appl. Math. 68 (12) (2015) 2108–2143) and (Large deviations and mixing for dissipative PDE’s with unbounded random kicks (2014) Preprint) in the case of kick forced dissipative PDE’s with parabolic regularization property such as, for example, the Navier–Stokes system and the complex Ginzburg–Landau equations. We also show that a high concentration towards the stationary measure is impossible, by proving that the rate function that governs the LDP cannot have the trivial form (i.e., vanish on the stationary measure and be infinite elsewhere).

A Local Large Deviation Principle for Inhomogeneous Birth–Death Processes

The paper considers a continuous-time birth–death process where the jump rate has an asymptotically polynomial dependence on the process position. We obtain a rough exponential asymptotic for the probability of trajectories of a re-scaled process contained within a neighborhood of a given continuous nonnegative function.

On the large deviation principle for maximum likelihood estimator of [formula omitted]-Brownian bridge

We consider the large deviation principle for maximum likelihood estimator of α-Brownian bridge. The full large deviation with explicit rate function is obtained in the case of non-Gaussian limit distribution by local large deviation principle and exponential tightness.

A Local Large Deviation Principle for Inhomogeneous Birth-Death Processes

The paper considers a continuous-time birth-death process where the jump rate has an asymptotically polynomial dependence on the process position. We obtain a rough exponential asymptotics for the probability of excursions of a re-scaled process contained within a neighborhood of a given continuous non-negative function.

LOCAL LARGE DEVIATIONS: McMILLIAN THEOREM FOR MULTITYPE GALTON-WATSON PROCESS

In this article we prove a local large deviation principle (LLDP) for the critical multitype Galton-Watson process from spectral potential point. We define the so-called a spectral potential $U_{\skrik}(\,\cdot,\,\pi)$ for the Galton-Watson process, where $\pi$ is the normalized eigen vector corresponding to the leading \emph{Perron-Frobenius eigen value } $\1$ of the transition matrix $\skria(\cdot,\,\cdot)$ defined from ${\skrik},$ the transition kernel. We show that the Kullback action or the deviation function, $J(\pi,\rho),$ with respect to an empirical offspring measure, $\rho,$ is the Legendre dual of $U_{\skrik}(\,\cdot,\,\pi).$ From the LLDP we deduce a conditional large deviation principle and a weak variant of the classical McMillian Theorem for the multitype Galton-Watson process. To be specific, given any empirical offspring measure $\varpi,$ we show that the number of critical multitype Galton-Watson processes on $n$ vertices is approximately $e^{n\langle \skrih_{\varpi},\,\pi\rangle},$ where $\skrih_{\varpi}$ is a suitably defined entropy.

Effects of Bed Forms and Large Protruding Grains on Near‐Bed Flow Hydraulics in Low Relative Submergence Conditions

AbstractIn mountain rivers, bed forms, large relatively immobile grains, and bed texture and topographic variability can significantly alter local and reach‐averaged flow characteristics. The low relative submergence of large immobile grains causes highly three‐dimensional flow fields that may not be represented by traditional shear stress, flow velocity, and turbulence intensity equations. To explore the influence of large protruding grains and bed forms on flow properties, we conducted a set of experiments in which we varied the relative submergence while holding the sediment transport capacity and upstream sediment supply constant. Flow and bed measurements were conducted at the beginning and end of each experiment to account for the absence or presence of bed forms, respectively. Detailed information on the flow was obtained by combining our measurements with a 3‐D numerical model. Commonly used velocity profile equations only performed well at the reach scale when shallow flow effects and the roughness length of the relatively mobile sediment were considered. However, at the local scale large deviations from these profiles were observed and simple methods to estimate the spatial distribution of near‐bed shear stresses are likely to be inaccurate. Zones of high turbulent kinetic energy occurred near the water surface and were largely controlled by the immobile grains and plunging flow. The reach‐averaged shear stress did not correlate to depth or slope, as commonly assumed, but instead was controlled by the relative boulder submergence and degree of plunging flow. For accurate flow predictions in mountain rivers, the effects of bed forms and large boulders must be considered.