Nizar Demni, Marguerite Zani
In this paper, we derive a handable expression for the Jacobi process semi group which is given by a bilinear series involving Jacobi polynomials. Our attempt uses a subordination of the considered process by means of a suitable random change. Once we did, we will be able, in the ultraspheric case, to derive a LDP for a family of estimators based on a single trajectory of the process.
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We continue to explore the connections between large deviations for objects coming from random matrix theory and sum rules. This connection was established in [17] for spectral measures of classical ensembles (Gauss-Hermite, Laguerre, Jacobi) and it was extended to spectral matrix measures of the Hermite and Laguerre ensemble in [20]. In this paper, we consider the remaining case of spectral matrix measures of the Jacobi ensemble. Our main results are a large deviation principle for such measures and a sum rule for matrix measures with reference measure the Kesten-McKay law. As an important in…
We establish uniqueness for a class of first-order Hamilton-Jacobi equations with Hamiltonians that arise from the large deviations of the empirical measure and empirical flux pair of weakly interacting Markov jump processes. As a corollary we obtain a large deviation principle for the trajectory of the empirical measure and empirical flux pair of such processes. As a second corollary we get same result in the setting where the jump-rates are time-periodic, with period-length that decreases to 0.
In this paper we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension $n$. In these cases the limit measure is given by the Marchenko-Pastur law and the semicircle law, respectively. For the weighted spectral measure we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.
We prove Freidlin-Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth-death processes, Galton-Watson trees, epidemic SI models, and prey-predator models. The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton-Jacobi equations. The notable feature for these Hamilton-Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton-Jacobi equations corresponding…
We study the Hamilton-Jacobi equation f - lambda Hf = h, where H f = e^{-f}Ae^f and where A is an operator that corresponds to a well-posed martingale problem. We identify an operator that gives viscosity solutions to the Hamilton-Jacobi equation, and which can therefore be interpreted as the resolvent of H. The operator is given in terms of optimization problem where the running cost is a path-space relative entropy. Finally, we use the resolvents to give a new proof of the abstract large deviation result of Feng and Kurtz.
We prove pathwise large-deviation principles of switching Markov processes by exploiting the connection to associated Hamilton-Jacobi equations, following Jin Feng's and Thomas Kurtz's method. In the limit that we consider, we show how the large-deviation problem in path-space reduces to a spectral problem of finding principal eigenvalues. The large-deviation rate functions are given in action-integral form. As an application, we demonstrate how macroscopic transport properties of stochastic models of molecular motors can be deduced from an associated principal-eigenvalue problem. The precise …
Nonstandard limit theorems and large deviation for beta -Jacobi ensembles with a different scaling
本文研究了参数为 $p_1, p_2\geq n$ 的 $β$-Jacobi 系综，通过谱测度方法证明了重缩放经验测度弱收敛到修正的 Watcher 律，并给出了相应重缩放谱测度的中心极限定理与大偏差。
Large deviation for Markov processes can be studied by Hamilton--Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton--Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) holds for the limit equation. Finally, we verify an exponential compact containment estimate. The large deviation principle then follows from the above three verifications. This paper illustrates such a m…
We prove the large deviation principle for the supports of Jacobi ensembles following Guionnet's method.
We prove the large deviation principle for the trajectory of a broad class of mean field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates. The main step in the proof of the large deviation principle, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations. Additionally, we show that the large deviation principle provides a general method to identify a Lyapunov …
Asymptotically efficient estimators for tail probabilities of extremals of $β$-Jacobi ensembles
本文研究 $β$-Jacobi 系综极值的尾部概率，通过对系综应用拉伸与平移变换确保极值满足大偏差，构造出依赖于大偏差速率函数的估计量，刻画了不同假设下极值的大偏差与中偏差行为。
State-Constrained Chemical Reactions: Discrete-to-Continuous Hamilton--Jacobi Equations and Large Deviations
本文研究离散状态空间上化学反应的宏观行为，通过WKB方法导出离散Hamilton--Jacobi方程，证明其解收敛于连续方程，并建立大偏差原理。
Large deviations for Markov processes with switching and homogenisation via Hamilton-Jacobi-Bellman equations
本文针对带内状态切换的分子马达扩散模型，通过分析同时依赖位置与动量的Hamilton–Jacobi–Bellman方程，用Feng-Kurtz方法证明了齐次化极限下轨迹的大偏差原理。
This is an expository note on large deviations, Hamilton-Jacobi-Bellman (HJB) equations, and the role of the Freidlin-Wentzell quasipotential in Chemical Reaction Networks (CRNs). The note was motivated by observations which identified Lyapunov functions for deterministic descriptions of CRNs by taking appropriate scaling limits of invariant distributions for the corresponding stochastic dynamics. We explain how this is a special case of a classical theory due to Freidlin and Wentzell. We also show that this Lyapunov function is a solution to the HJB partial differential equation if and only i…
Viscosity solutions of Hamilton-Jacobi equation in $RCD(K,\infty)$ spaces and applications to large deviations
本文在RCD(K,∞)空间中推导Hamilton-Jacobi方程粘性逼近解的梯度与拉普拉斯收缩估计，证明粘性趋于零时解收敛于Hopf–Lax演化，进而研究热核与布朗运动的短时大偏差原理，并讨论Schrödinger问题到二次最优输运的Γ收敛。
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.
We study the global fluctuations for linear statistics of the form $\sum_{i=1}^n f(λ_i)$ as $n \rightarrow \infty$, for $C^1$ functions $f$, and $λ_1, ..., λ_n$ being the eigenvalues of a (general) $β$-Jacobi ensemble, for which tridiagonal models were given by Killip and Nenciu as well as Edelman and Sutton. The fluctuation from the mean ($\sum_{i=1}^n f(λ_i) - \Exp \sum_{i=1}^n f(λ_i)$) is given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation …
A sum rule is an identity connecting the entropy of a measure with coefficients involved in the construction of its orthogonal polynomials (Jacobi coefficients). Our paper is an extension of Gamboa, Nagel and Rouault (2016), where we have showed sum rules by using only probabilistic tools (namely the large deviations theory). Here, we prove large deviation principles for the weighted spectral measure of unitarily invariant random matrices in two general situations: firstly, when the equilibrium measure is not necessarily supported by a single interval and secondly, when the potential is a nonn…
We establish Cramér type moderate deviation (MD}) results for heavy trimmed L-statistics; we obtain our results under a very mild smoothness condition on the inversion $F^{-1}$ ($F$ is the underlying distribution of i.i.d. observations) near two points, where trimming occurs, we assume also some smoothness of weights of the L-statistic. Our results complement previous work on Cramér type large deviations (LD) for trimmed L-statistics by Gribkova (2016) and Callaert et al. (1982).
In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed $L$-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with ones in Callaert et al.(1982) -- the first and, as far as we know, the single article, where some results on probabilities of large deviations for the trimmed $L$-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed $L$-statistic by a non-trimmed $L$-statistic (with smooth weight function) based on Winsorized random variables. U…
Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Langrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation $dX_t=b(X_t)dt+εdW_t$ where $W_t$ is a Brown…
We present stochastic homogenization results for viscous Hamilton-Jacobi equations using a new argument which is based only on the subadditive structure of maximal subsolutions (solutions of the "metric problem"). This permits us to give qualitative homogenization results under very general hypotheses: in particular, we treat non-uniformly coercive Hamiltonians which satisfy instead a weaker averaging condition. As an application, we derive a general quenched large deviations principle for diffusions in random environments and with absorbing random potentials.
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 …
Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an estimate for the characteristic functions of the linear statistics; this estimate is uniform over the growing interval. The proof of the lemma relies on the Riemann--Hilbert approach.
Cramér-type moderate deviation for double index permutation statistics
本文提出双索引排列统计的Cramér型中等偏差定理，解决了传统方法无法捕捉稀疏统计量最优收敛速率的问题，扩展至稀疏统计量如下降数和Chatterjee秩相关系数。
We consider a multinomial distribution, where the number of cells increases and the cell-probabilities decreases as the number of observations grows. The probabilities of large deviations of statistics, which has form of a sum of Borel functions of cell-frequencies, are studied. Large deviation results for the power-divergence statistics and its most popular special variants, as well as for some count statistics are derived as consequences of general theorems.
Let $(ξ_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$ \mathbf{P} \Big( \max_{1\leq k \leq n} \sum_{i=1}^{k} |ξ_i|\big/ \big(\sum_{i=1}^{n} |ξ_i|^β\big)^{1/β} \geq x \Big) $$ for $x>0$ and $β>1.$ Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. An application to Student's $t$-statistics is also given.
This paper is devoted to the problem of sample path large deviations for the Markov processes on R_+^N having a constant but different transition mechanism on each boundary set {x:x_i=0 for i\notinΛ, x_i>0 for i\inΛ}. The global sample path large deviation principle and an integral representation of the rate function are derived from local large deviation estimates. Our results complete the proof of Dupuis and Ellis of the sample path large deviation principle for Markov processes describing a general class of queueing networks.
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…
Heavy tailed large deviations for time averages of diffusions: the Ornstein-Uhlenbeck case
本文研究 Ornstein-Uhlenbeck 过程任意幂次时间平均值的大偏差，证明超过临界值后偏差呈次指数衰减且速率函数非凸，并给出简短自洽的证明。
The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erdős-Rényi random graphs and $U$-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices…
We consider two-dimensional determinantal processes which are rotation-invariant and study the fluctuations of the number of points in disks. Based on the theory of mod-phi convergence, we obtain Berry-Esseen as well as precise moderate to large deviation estimates for these statistics. These results are consistent with the Coulomb gas heuristic from the physics literature. We also obtain functional limit theorems for the stochastic process $(\# D_r)_{r>0}$ when the radius $r$ of the disk $D_r$ is growing in different regimes. We present several applications to invariant determinantal process,…
We 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 towards this l…
Moderate deviation principle for the chi-square statistics
本文研究样本量增大时有限字母表动态变化下的Pearson卡方统计量，建立了其中偏差原理。
For arbitrary $β> 0$, we use the orthogonal polynomials techniques developed by R. Killip and I. Nenciu to study certain linear statistics associated with the circular and Jacobi $β$ ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.
A new large deviation results for the Pearson chi-square and Log-likelihood ratio statistics are obtained. Here attention is focused on the case when the number of groups increases to infinity and the probabilities of groups decreases to zero, as the sample size tends to infinity.
Deviation and moment inequalities for Banach-valued $U$-statistics
本文在可分 Banach 空间光滑性假设下建立了独立数据 U 统计量的偏差不等式，并将其用于大数定律速率、Hölder 型泛函中心极限定理和不完全 U 统计量的矩不等式。
Large deviation properties for pattern statistics in primitive rational models
本研究针对本原有理随机模型下的模式统计量，推导了依赖转移矩阵主特征值与特征向量的大偏差速率函数，并在温和条件下将其有效区间扩展至最大开区间(0,1)。
Two-sample $U$-statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cramér-type moderate deviation theorems for Studentized two-sample $U$-statistics in a general framework, including the two-sample $t$-statistic and Studentized Mann-Whitney test statistic as prototypical examples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample $t$-statistic. These results extend the applicability of the existing statistical methodologies f…
In this paper we study the large deviations of time averaged mean square displacement (TAMSD) for Gaussian processes. The theory of large deviations is related to the exponential decay of probabilities of large fluctuations in random systems. From the mathematical point of view a given statistics satisfies the large deviation principle, if the probability that it belongs to a certain range decreases exponentially. The TAMSD is one of the main statistics used in the problem of anomalous diffusion detection. Applying the theory of generalized chi-squared distribution and sub-gamma random variabl…
Define the incremental fractional Brownian field $B_{H}(s+τ)-B_{H}(s), H\in (0,1)$, where $B_{H}(s)$ is a standard fractional Brownian motion with Hurst index $H\in(0,1)$. In this paper we derive the exact asymptotic behaviour of the maximum $\max_{(τ,s)\in[0,1]\times[0,T]} (B_{H}(s+τ)-B_{H}(s)) $ for any $H\in (0,1/2)$ complimenting thus the result of Zholud (2008) for the Brownian motion.
Precise large deviations in geometric last passage percolation
本文研究几何 LPP 的最后通过时间，通过建立与 JUE 和 TUE 最大特征值相关的两个对偶公式，精确刻画上下尾大偏差概率至常数项，并得到 TUE 特征多项式矩的渐近性。
Cramér type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cramér type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized $U$-statistics.
We study deviation of U-statistics when samples have heavy-tailed distribution so the kernel of the U-statistic does not have bounded exponential moments at any positive point. We obtain an exponential upper bound for the tail of the U-statistics which clearly denotes two regions of tail decay, the first is a Gaussian decay and the second behaves like the tail of the kernel. For several common U-statistics, we also show the upper bound has the right rate of decay as well as sharp constants by obtaining rough logarithmic limits which in turn can be used to develop LDP for U-statistics. In spite…
Self-normalized Cramér type moderate deviations for martingales and applications
本文在温和条件下建立了鞅的自归一化Cramér型中等偏差，推广了Fan等人2019年的工作，并将其应用于Student统计量、平稳鞅差序列及随机环境中的分支过程，特别给出了后者t-统计量的相应结果。
A two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged regular closed region K whose charge density is determined by its equilibrium potential at an inverse temperature βis investigated. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a point process in the complex plane. We describe the weak* limits of the joint intensities of this point process and show that it is exponentially likely to find the process in a neighborhood of the equ…
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