Uwe Einmahl, James Kuelbs
Precise asymptotics for moderate deviation probabilities are established for open convex sets in both the finite- and infinite-dimensional settings. Our results are based on the existence of dominating points for these sets, a related representation formula, and asymptotics for the integral term in this formula.
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Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain ``clumps'' around high-level crossings. We also discuss how these results are related t…
Upper moderate deviation probabilities for the maximum of branching Brownian motion
本文研究分支布朗运动极大值的上中等偏差概率，利用修正的第二矩方法给出其渐近等价，并刻画了贡献粒子的典型行为。
Given a super-critical branching random walk on $\mathbb R$ started from the origin, let $M_n$ be the maximal position of individuals at the $n$-th generation. Under some mild conditions, it is known from \cite{A13} that as $n\rightarrow\infty$, $M_n-x^*n+\frac{3}{2θ^*}\log n$ converges in law for some suitable constants $x^*$ and $θ^*$. In this work, we investigate its moderate deviation, in other words, the convergence rates of $$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2θ^*}\log n-\ell_n\right),$$ for any positive sequence $(\ell_n)$ such that $\ell_n=O(n)$ and $\ell_n\uparrow\infty$. As a by-…
In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the us…
Some results on probabilities of moderate deviations
本文研究了独立同分布随机变量序列的中等偏差概率，在 $g(\log n)$ 尺度上获得了精确的渐近估计，并证明了该结果同时依赖于方差和尾分布的渐近行为。
Deviation and moderate deviation for extremal eigenvalues of large Chiral non-Hermitian random matrices
本文研究了带参数 $n$ 和 $v$ 的大型手性非厄米随机矩阵系综，建立了其谱半径及相关极值特征值平方的偏差概率与中偏差概率。
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.
The purpose of the present paper is to establish explicit bounds on moderate deviation probabilities for a rather general class of geometric functionals enjoying the stabilization property, under Poisson input and the assumption of a certain control over the growth of the moments of the functional and its radius of stabilization. Our proof techniques rely on cumulant expansions and cluster measures and yield completely explicit bounds on deviation probabilities. In addition, we establish a new criterion for the limiting variance to be non-degenerate. Moreover, our main result provides a new ce…
Upper moderate deviation probabilities for the maximum of a branching random walk
本文研究超临界分支随机游走最大值的上适度偏差概率，基于Bramson等人策略，得到渐近等价并分析粒子行为。
Poisson approximation using Stein's method has been extensively studied in the literature. The main focus has been on bounding the total variation distance. This paper is a first attempt on moderate deviations in Poisson approximation for right-tail probabilities of sums of dependent indicators. We obtain results under certain general conditions for local dependence as well as for size-bias coupling. These results are then applied to independent indicators, 2-runs, and the matching problem.
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…
We present precise moderate deviation probabilities, in both quenched and annealed settings, for a recurrent diffusion process with a Brownian potential. Our method relies on fine tools in stochastic calculus, including Kotani's lemma and Lamperti's representation for exponential functionals. In particular, our result for quenched moderate deviations is in agreement with a recent theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571-609] who studied the corresponding problem for Sinai's random walk in random environment.
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…
In the present paper we consider the fluctuations of the free energy in the random energy model (REM) on a moderate deviation scale. We find that for high temperatures the normal approximation holds only in a narrow range of scalings away from the CLT. For scalings of higher order, probabilities of moderate deviations decay faster than exponentially.
Large and moderate deviation principles for stochastic partial differential equation on graph
本文针对度量图上随机偏微分方程特有挑战，引入图上新型加权Sobolev空间与适配紧嵌入结果，通过弱收敛方法研究其大、中偏差原理，为相关多尺度模型提供新的偏差结果。
Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices
研究椭圆变形实Ginibre集合中实特征值数量的统计特性，推导中等至大偏差情形下稀有事件的概率，连接高斯波动与大偏差理论。
The deviation principles of record numbers in random walk models have not been completely investigated, especially for the non-nearest neighbor cases. In this paper, we derive the asymptotic probabilities of large and moderate deviations for the number of "weak records"(or "ladder points") in two kinds of one-dimensional non-nearest neighbor random walks. The proofs depend only on the direct analysis of random walks. We illustrate that the traditional method of analyzing the local time of Brownian motions, which is often adopted for the simple random walks, would lead to wrong conjectures for …
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。
The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by …
Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime
本文提出一种可证明高效的重要性采样方案，用于估计小噪声随机反应扩散方程的解从稳定平衡点标定邻域退出的概率，通过中等偏差缩放和线性化近似，并结合随机控制与变分方法证明了其在零噪声极限和预渐近情况下的良好表现。
In this paper, using techniques developed in our earlier works on the theory of mod-Gaussian convergence, we prove precise moderate and large deviation results for the logarithm of the characteristic polynomial of a random unitary matrix. In the case where the unitary matrix is chosen according to the Haar measure, the logarithms of the probabilities of fluctuations of order $A=O(N)$ of the logarithm of the characteristic polynomial have been estimated by Hughes, Keating and O'Connell. In this work we give an equivalent of the probabilities themselves (without the logarithms), and we do so for…
Large and Moderate Deviations for Conservative Tail-Index Estimation
针对罕见事件概率估计中尾指数高估问题，推导Hill估计器的大偏差和中偏差渐近，构造偏差与衰减速率最优平衡的估计器，并分析偏差校正及边界条件。
We prove precise deviations results in the sense of Cramér and Petrov for the upper tail of the distribution of the maximal value for a special class of determinantal point processes that play an important role in random matrix theory. Here we cover all three regimes of moderate, large and superlarge deviations for which we determine the leading order description of the tail probabilities. As a corollary of our results we identify the region within the regime of moderate deviations for which the limiting Tracy-Widom law still predicts the correct leading order behavior. Our proofs use that the…
Moment generating functions and moderate deviation principles for lacunary trigonometric sums
本文研究缺项三角和的矩生成函数，确定算术效应起作用的精确阈值，证明其中偏差原理，证实尾概率在中心极限定理与大偏差原理区间内符合高斯行为。
Deviation probabilities and Sharp Berry-Esseen bound for rightmost eigenvalue of large non-Hermitian chiral random matrices
本文研究了大非厄密奇随机矩阵最右特征值的偏差概率，得出了精确的伯恩-埃森界，并给出了大和中等偏差原理。
Tail Bounds for Queues with Abandonment: Constant, Moderate, Large Deviations, and Efficient Concentration
研究高负载单服务器队列中放弃现象的尾概率界，通过Stein方法和变换方法获得不同偏差范围下的高效集中界，包括常数偏差的高斯衰减、中等偏差的亚高斯尾和大偏差的亚泊松衰减。
The stochastic six vertex model and discrete orthogonal polynomial ensembles
研究随机六顶点模型中高度函数的中等偏差估计，通过分析离散正交多项式族的乘法统计特性，揭示其在不同缩放下的普遍交叉行为。
Geometric sharp large deviations for random projections of $\ell_p^n$ spheres and balls
本文研究高维 ℓ_p^n 球面和球体随机标量投影的尾概率，建立淬火尖锐大偏差估计，揭示前因子对投影方向的几何依赖，并通过数值方法验证其中等维数近似效果。
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