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Explicit bivariate rate functions for large deviations in AR(1)and MA(1)processes with Gaussian innovations

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Probability, Uncertainty and Quantitative Risk 2023年第2期8卷 177-212页

作者： Maicon J.Karling Artur O.Lopes Sílvia R.C.Lopes Mathematics and Statistics Institute Federal University of Rio Grande do Sul91500-900 Porto AlegreRSBrazil

We investigate the large deviations properties for centered stationary AR(1)and MA(1)processes with independent Gaussian innovations,by giving the explicit bivariate rate functions for the sequence of two-dimensional random vectors.Via the Contraction Principle,we provide the explicit rate functions for the sample mean and the sample second moment.In the AR(1)case,we also give the explicit rate function for the sequence of two-dimensional random vectors(W_(n))n≥2=(n^(-1(∑_(k=1)^(n)X_(k),∑_(k=1)^(n)X_(k)^(2))))_(n∈N)n≥2,but we obtain an analytic rate function that gives different values for the upper and lower bounds,depending on the evaluated set and its intersection with the respective set of exposed points.A careful analysis of the properties of a certain family of Toeplitz matrices is necessary.The large deviations properties of three particular sequences of one-dimensional random variables will follow after we show how to apply a weaker version of the Contraction Principle for our setting,providing new proofs for two already known results on the explicit deviation function for the sample second moment and Yule-Walker estimators.We exhibit the properties of the large deviations of the first-order empirical autocovariance,its explicit deviation function and this is also a new result.

关键词： Autoregressive processes Empirical autocovariance large deviations Moving average processes Sample moments Toeplitz matrices Yule-Walker estimator

large deviations and Moderate deviations for Kernel Density Estimators of Directional Data

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Acta Mathematica Sinica,English Series 2010年第5期26卷 937-950页

作者： Fu Qing GAO Li Na LI School of Mathematics and Statistics Wuhan University Wuhan 430072 P. R. China Department of Mathematics Tongji University Shanghai 200092 P. R. China

Let fn be the non-parametric kernel density estimator of directional data based on a kernel function K and a sequence of independent and identically distributed random variables taking values in d-dimensional unit sphere Sd-1. It is proved that if the kernel function is a function with bounded variation and the density function f of the random variables is continuous, then large deviation principle and moderate deviation principle for {sup x∈sd-1 ｜fn（x） - E（fn（x））｜, n ≥ 1} hold.

关键词： kernel density estimator directional data moderate deviations large deviations

large deviations AND MODERATE deviations FOR m-NEGATIVELY ASSOCIATED RANDOM VARIABLES

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Acta Mathematica Scientia 2007年第4期27卷 886-896页

作者：胡亦钧明瑞星杨文权 School of Mathematics and Statistics Wuhan University Wuhan 430072China School of Mathematics and Statistics Wuhan University Wuhan 430072China Department of MathematicsJianghran UniversityWuhan 430056China

M-negatively associated random variables, which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its particular case, are introduced and studied. large deviation principles and moderate deviation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.

关键词： negatively associated random variables stationary sequence strong law of large numbers large deviations moderate deviations

large deviations for Hitting Times of a Random Walk in Random Environment on a Strip

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Acta Mathematica Sinica,English Series 2014年第3期30卷 395-410页

作者： Mei Juan ZHANG School of Statistics and Mathematics Central University of Finance and Economics

We consider a random walk in random environment on a strip, which is transient to the right. The random environment is stationary and ergodic. By the constructed enlarged random environment which was first introduced by Goldsheid （2008）, we obtain the large deviations conditioned on the environment （in the quenched case） for the hitting times of the random walk.

关键词： Random walk in random environment strip large deviations quenched enlarged randomenvironment

large deviations in Generalized Random Graphs with Node Weights

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Acta Mathematica Sinica,English Series 2018年第10期34卷 1626-1634页

作者： Qun LIU Zhi Shan DONG Mathematics School and Institute of Jilin University

Generalized random graphs are considered where the presence or absence of an edge de- pends on the weights of its nodes. Our main interest is to investigate large deviations for the number of edges per node in such a generalized random graph, where the node weights are deterministic under some regularity conditions, as well as chosen i.i.d, from a finite set with positive components. When the node weights are random variables, obstacles arise because the independence among edges no longer exists, our main tools are some results of large deviations for mixtures. After calculating, our results show that the corresponding rate functions for the deterministic case and the random case are very different.

关键词： large deviations mixture generalized random graphs

large deviations for Parameter Estimators of Some Time Inhomogeneous Diffusion Process

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Acta Mathematica Sinica,English Series 2011年第11期27卷 2245-2258页

作者： Shou Jiang ZHAO Fu Qing GAO School of Mathematics and Statistics Wuhan University Wuhan 430072 P. R. China School of Science China Three Gorges University Yichang 443002 P. R. China

The goal of this paper is to study large deviations for estimator and score function of some time inhomogeneous diffusion process. large deviation in the non-steepness case with explicit rate functions is obtained by ... 详细信息

关键词： Parameter estimation large deviations time inhomogeneous diffusion process

large deviations for Empirical Measures of Not Necessarily Irreducible Countable Markov Chains with Arbitrary Initial Measures

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Acta Mathematica Sinica,English Series 2005年第6期21卷 1377-1390页

作者： Yi Wen JIANG Li Ming WU Department of Mathematics Military Economics Academy Wuhan 430035 P. R. China Laboratoire de Mathématiques Appliqudes CNRS UMR 6620 Universitd Blaise Pascal 63177 Aubière France Department Mathematics Wuhan University Wuhan 430072 P. R. China

All known results on large deviations of occupation measures of Markov processes are based on the assumption of （essential） irreducibility. In this paper we establish the weak＊ large deviation principle of occupation measures for any countable Markov chain with arbitrary initial measures. The new rate function that we obtain is not convex and depends on the initial measure, contrary to the （essentially） irreducible case.

关键词： large deviations Markov processes （chains）

Finite Time Ruin Probabilities and large deviations for Generalized Compound Binomial Risk Models

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Acta Mathematica Sinica,English Series 2005年第5期21卷 1099-1106页

作者： Yi Jun HU School of Mathematics and Statistics Wuhan University Wuhan 430072 P. R. China

In this paper, we extend the classical compound binomial risk model to the case where the premium income process is based on a Poisson process, and is no longer a linear function. For this more realistic risk model, Lundberg type limiting results for the finite time ruin probabilities are derived. Asymptotic behavior of the tail probabilities of the claim surplus process is also investigated.

关键词： Ruin probability （Generalized） compound binomial risk model large deviations

large deviations for Heavy-tailed Random Variables in Prospective-loss Process

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Communications in Mathematical Research 2009年第3期25卷 223-230页

作者： BAO ZHEN-HUA School of Mathematics Liaoning Normal University Dalian 116029

In this paper, we study the precise large deviations for the prospectiveloss process with consistently varying tails. The obtained results improve some related known ones.

关键词： consistently varying tails large deviations prospective-loss process