Consider the state space model (X_t,Y_t), where (X_t) is a Markov chain, and (Y_t) are the observations. In order to solve the so-called filtering problem, one has to compute L(X_t|Y_1,...,Y_t), the law of X_t given the observations (Y_1,...,Y_t). The particle filtering method gives an approximation of the law L(X_t|Y_1,...,Y_t) by an empirical measure \frac{1}{n}\sum_1^nδ_{x_{i,t}}. In this paper we establish the moderate deviation principle for the empirical mean \frac{1}{n}\sum_1^nψ(x_{i,t}) (centered and properly rescaled) when the number of particles grows to infinity, enhancing the central limit theorem. Several extensions and examples are also studied.
The purpose of this paper is to investigate the deviation inequalities and the moderate deviation principle of the least squares estimators of the unknown parameters of general $p$th-order bifurcating autoregressive processes, under suitable assumptions on the driven noise of the process. Our investigation relies on the moderate deviation principle for martingales.
We establish the moderate deviation principle for the solutions of a class of stochastic partial differential equations with non-Lipschitz continuous coefficients. As an application, we derive the moderate deviation principle for two important population models: super-Brownian motion and Fleming-Viot process.
In this paper, we prove a central limit theorem and establish a moderate deviation principle for the the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity. The proof for moderate deviation principle is based on the weak convergence approach.
The term "moderate deviations" is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about "non-central moderate deviations" when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps.
We apply Lindeberg's method, invented to prove a central limit theorem, to analyze the moderate deviations around such a central limit theorem. In particular, we will show moderate deviation principles for martingales as well as for random sums, in the latter situation both, in the case when the limit distribution is Gaussian or non-Gaussian. Moreover in the Gaussian case we show moderate deviations for random sums using bounds on cumulants, alternatively. Finally, we also prove a large deviation principle for certain random sums.
In this paper, we establish a moderate deviations principle for the Langevin dynamics with strong damping. The weak convergence approach plays an important role in the proof.
In this paper, we establish a central limit theorem and a moderate deviations for 2D stochastic primitive equations with multiplicative noise. The proof is mainly based on the weak convergence approach.
An example of a discrete-time stationary random process whose sums follow the normal approximation within a given part of the region of moderate deviations, but violate it outside this part.
In this paper, we present sufficient conditions and criteria to establish the large and moderate deviation principle of multivalued McKean-Vlasov stochastic differential equation by means of the weak convergence method.
In this paper we are concerned with hitting times of a family of density-dependent Markov chains. A moderate deviation principle of the hitting time is given. The proof of the main theorem relies heavily on moderate deviations of density-dependent Markov chains given in \cite{Xue2021} and upper bounds of large deviations of Markov processes given in \cite{Dupuis1991}. An analogue moderate deviation of the hitting time of the diffusion approximation of the density-dependent Markov chain introduced in \cite{Ethier1986} is also given.
In Stein's method, the exchangeable pair approach is commonly used to estimate the approximation errors in normal approximation. In this paper, we establish a Cramér-type moderate deviation theorem of normal approximation for unbounded exchangeable pairs. As applications, Cramér-type moderate deviation theorems for the sums of local statistics and general Curie--Weiss model are obtained.
A uniform key renewal theorem is deduced from the uniform Blackwell's renewal theorem. A uniform LDP (large deviations principle) for renewal-reward processes is obtained, and MDP (moderate deviations principle) is deduced under conditions much weaker than existence of exponential moments.
We consider the one dimensional symmetric simple exclusion process with a slow bond. In this model, particles cross each bond at rate $N^2$, except one particular bond, the slow bond, where the rate is $N$. Above, $N$ is the scaling parameter. This model has been considered in the context of hydrodynamic limits, fluctuations and large deviations. We investigate moderate deviations from hydrodynamics and obtain a moderate deviation principle.
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. The results can be applied to the design of accelerated Monte Carlo algorithms for certain problems, where schemes based on moderate deviations are easier to construct and in certain situations provide performance comparable to those based on large deviations.
By comparing the original equations with the corresponding stationary ones, the moderate deviation principle (MDP) is established for unbounded additive functionals of several different models of distribution dependent SDEs, with non-degenerate and degenerate noises.
In this paper, we obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extend the results of Chung, Kennedy, Kaigh and Khorunzhiy and Marckert.
We establish a moderate deviation principle for processes with independent increments under certain growth conditions for the characteristics of the process. Using this moderate deviation principle, we give a new proof for Strassen's functional law of the iterated logarithm. In particular, we show that any square-integrable Lévy process satisfies Strassen's law.
We prove a moderate deviations principles for the size of the largest connected component in a random $d$-uniform hypergraph. The key tool is a version of the exploration process, that is also used to investigate the giant component of an Erdös-Rényi graph, a moderate deviations principle for the martingale associated with this exploration process, and exponential estimates.
A moderate deviation principle for nonlinear functions of Gaussian processes is established. The nonlinear functions need not be locally bounded. Especially, the logarithm is allowed. (Thus, small deviations of the process are relevant.) Both discrete and continuous time is treated. An integrable power-like decay of the correlation function is assumed.
We prove large and moderate deviation principles for the distribution of an empirical mean conditioned by the value of the sum of discrete i.i.d. random variables. Some applications for combinatoric problems are discussed.
This article is concerned with moderate deviation principles of a general class of mean eld type interacting particle models. We discuss functional moderate deviations of the occupation measures for both the strong -topology on the space of fi nite and bounded measures as well as for the corresponding stochastic processes on some class of functions equipped with the uniform topology. Our approach is based on an original semigroup analysis combined with stochastic perturbation techniques and projective limit large deviation methods.
We derive moderate deviation principles for the trajectory of the empirical magnetization of the standard Curie-Weiss model via a general analytic approach based on convergence of generators and uniqueness of viscosity solutions for associated Hamilton-Jacobi equations. The moderate asymptotics depend crucially on the phase under consideration.
Using martingale methods, we obtain some upper bounds for large and moderate deviations of products of independent and identically distributed elements of GL d (R). We investigate all the possible moment conditions, from super-exponential moments to weak moments of order p \textgreater{} 1, to get a complete picture of the situation. We also prove a moderate deviation principle under an appropriate tail condition.
In this paper, we are concerned with SIR epidemics in a random environment on complete graphs, where every edges are assigned with i.i.d. weights. Our main results give large and moderate deviation principles of sample paths of this model.
We prove moderate deviation principles for the tagged particle position and current in one-dimensional symmetric simple exclusion processes. There is at most one particle per site. A particle jumps to one of its two neighbors at rate $1/2$, and the jump is suppressed if there is already one at the target site. We distinguish one particular particle which is called the tagged particle. We first establish a variational formula for the moderate deviation rate functions of the tagged particle positions based on moderate deviation principles from hydrodynamic limit proved by Gao and Quastel \cite{g…
In this paper we prove a central limit theorem and a moderate deviation principle for a class of semilinear stochastic partial differential equations, which contain Burgers' equation and the stochastic reaction-diffusion equation. The weak convergence method plays an important role.
In this paper, we prove a central limit theorem and estabilish a moderate deviation principle for stochastic models of incompressible second fluids. The weak convergence method inreoduced by [4] plays an important role.
Record numbers are basic statistics in random walks, whose deviation principles are not very clear so far. In this paper, the asymptotic probabilities of large and moderate deviations for numbers of weak records in right continuous or left continuous random walks are proved.
By extending the methods in Peligrad et al. (2014a, b), 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.
Motivated by the study of dependent random variables by coupling with independent blocks of variables, we obtain first sufficient conditions for the moderate deviation principle in its functional form for triangular arrays of independent random variables. Under some regularity assumptions our conditions are also necessary in the stationary case. The results are then applied to derive moderate deviation principles for linear processes, kernel estimators of a density and some classes of dependent random variables.
Moderate deviation principles for stochastic differential equations driven by a Poisson random measure (PRM) in finite and infinite dimensions are obtained. Proofs are based on a variational representation for expected values of positive functionals of a PRM.
The term \emph{moderate deviations} is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. In this paper we present some examples of classes of large deviation principles of this kind, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.
Given a symmetric random walk in $Z^2$ with finite second moments, let $R_n$ be the range of the random walk up to time $n$. We study moderate deviations for $R_n -E R_n$ and $E R_n -R_n$. We also derive the corresponding laws of the iterated logarithm.
We prove a moderate deviation principle for subgraph count statistics of Erdos-Renyi random graphs. This is equivalent in showing a moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an estimation of the log-Laplace transform and the Gaertner-Ellis theorem. We obtain upper bounds on the upper tail probabilities of the number of occurrences of small subgraphs. The method of proof is used to show supplemental moderate deviation principles for a class of symmetric statistics, including non-degenerate U-statistics with independent or Markovian entrie…
In this paper, we prove a central limit theorem and a moderate deviation principle for a perturbed stochastic Cahn-Hilliard equation defined on [0, T]x [0, π]^d, with d \in {1,2,3}. This equation is driven by a space-time white noise. The weak convergence approach plays an important role.
This paper is a further investigation of the generalized $N$-urn Ehrenfest model introduced in \cite{Xue2020}. A moderate deviation principle from the hydrodynamic limit of the model is derived. The proof of this main result follows a routine procedure introduced in \cite{Kipnis1989}, where a replacement lemma plays the key role. To prove the replacement lemma, the large deviation principle of the model given in \cite{Xue2020} is utilized.
A Cramér-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides theoretical justification when the limiting tail probability can be used to estimate the tail probability under study. Chen Fang and Shao (2013) obtained a general Cramér-type moderate result using Stein's method when the limiting was a normal distribution. In this paper, Cramér-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Stein's coupling. In particular…
In this paper we obtain a Bernstein type inequality for a class of weakly dependent and bounded random variables. The proofs lead to a moderate deviations principle for sums of bounded random variables with exponential decay of the strong mixing coeficients that complements the large deviation result obtained by Bryc and Dembo (1998) under superexponential mixing rates.
This is basically a polished presentation for Sections 1,2 of arXiv:0801.1050. The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. Here it is established for a new class of random processes. The approach is promising also for random fields.
We establish Cramér-type moderate deviation theorems for sums of locally dependent random variables and combinatorial central limit theorems. Under some mild exponential moment conditions, optimal error bounds and convergence ranges are obtained. Our main results are more general or shaper than the existing results in the literature. The main results follows from a more general Cramér-type moderate deviation theorem for dependent random variables without any boundedness assumptions, which is of independent interest. The proofs couple Stein's method with a recursive argument.
We establish a moderate deviations principle (MDP) for the log-determinant $\log | \det (M_n) |$ of a Wigner matrix $M_n$ matching four moments with either the GUE or GOE ensemble. Further we establish Cramér--type moderate deviations and Berry-Esseen bounds for the log-determinant for the GUE and GOE ensembles as well as for non-symmetric and non-Hermitian Gaussian random matrices (Ginibre ensembles), respectively.
In the present paper we prove moderate deviations for a Curie-Weiss model with external magnetic field generated by a dynamical system, as introduced by Dombry and Guillotin-Plantard. The results extend those already obtained in the case of a constant external field by Eichelsbacher and Löwe. The Curie-Weiss model with dynamic external field is related to the so called dynamic Z-random walks. We also prove a moderate deviation result for the dynamic Z-random walk, completing the list of limit theorems for this object.
The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. Here it is established for splittable random fields. Version 2. Adapted to new Version 3 of [2] arXiv:1706.00991. Terminology: "CMS random fields". Numbers of items in [2] updated. References: "response", not "responce". Otherwise unchanged.
The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. Here it is established for splittable random fields integrated against test functions. Version 2: minor changes. Interval [0,1] replaced with [0,1) for better compatibility with Part V. Copyedit pages 3, 4, 6, 16, 22, 23.
In this paper, we prove the moderate deviations principle (MDP) for a general system of slow-fast dynamics. We provide a unified approach, based on weak convergence ideas and stochastic control arguments, that cover both the averaging and the homogenization regimes. We allow the coefficients to be in the whole space and not just the torus and allow the noises driving the slow and fast processes to be correlated arbitrarily. Similar to the large deviation case, the methodology that we follow allows construction of provably efficient Monte Carlo methods for rare events that fall into the moderat…
In this paper, we establish a moderate deviation principle for stochastic models of two-dimensional second grade fluids driven by Lévy noise. We will adopt the weak convergence approach. Because of the appearance of jumps, this result is significantly different from that in Gaussian case.
The purpose of this paper is to investigate moderate deviations for the Durbin-Watson statistic associated with the stable first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We first establish a moderate deviation principle for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise. It enables us to provide a moderate deviation principle for the Durbin-Watson statistic in the easy case where the driven noise is normally …
We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, …
We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thäle (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.
In this paper, we establish a central limit theorem (CLT) and the moderate deviation principles (MDP) for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to stochastic partial differential equations of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.
In this paper, we proved a central limit theorem and established a moderate deviation principle for a perturbed stochastic wave equation defined on $[0,T]\times \rr^3$. This equation is driven by a Gaussian noise, white in time and correlated in space. The weak convergence approach plays an important role.
