摘要翻译：
我们给出了$m_n$的分布，它是一个由1阶移动平均的$n$观测值序列的最大值。首先用重复积分的形式给出了解，然后对下面的独立随机变量具有绝对连续密度的情况给出了解。当相关性为正时，$$P(M_n%\max^n_{i=1}X_i\leqx)=\\sum_{j=1}^\infty\beta_{jx}\nu_{jx}^{n}\\B_{x}\nu_{1x}^{n}$$其中%$\{X_i}$是具有正相关性的1阶移动平均值，$\nu_{jx}\$是Fredholm核的特征值（奇异值），$\nu_{1x}$是最大值的特征值。当相关性为负值时，也给出了类似的结果。结果类似于估计的大偏差扩展，因为最大值不需要标准化以有一个极限。在连续情况下，左、右本征函数的积分方程转化为一阶线性微分方程。对于给定矩阵的某些已知权值$\{w_i\}$和特征值$\{theta_i\}$,特征值满足形式为$$\sum_{i=1}^\infty w_i(\lambda-\theta_i)^{-1}=\lambda-\theta_0$$的方程。这可以通过将和截断为越来越多的项来解决。
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英文标题：
《The distribution of the maximum of a first order moving average: the
  continuous case》
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作者：
Christopher S. Withers and Saralees Nadarajah
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最新提交年份：
2009
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分类信息：

一级分类：Statistics        统计学
二级分类：Methodology        方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
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一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
  We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables have an absolutely continuous density. When the correlation is positive, $$ P(M_n %\max^n_{i=1} X_i \leq x) =\ \sum_{j=1}^\infty \beta_{jx} \nu_{jx}^{n} \approx B_{x} \nu_{1x}^{n} $$ where %$\{X_i\}$ is a moving average of order 1 with positive correlation, and $\{\nu_{jx}\}$ are the eigenvalues (singular values) of a Fredholm kernel and $\nu_{1x}$ is the eigenvalue of maximum magnitude. A similar result is given when the correlation is negative. The result is analogous to large deviations expansions for estimates, since the maximum need not be standardized to have a limit. % there are more terms, and $$P(M_n <x) \approx B'_{x}\ (1+\nu_{1x})^n.$$   For the continuous case the integral equations for the left and right eigenfunctions are converted to first order linear differential equations. The eigenvalues satisfy an equation of the form $$\sum_{i=1}^\infty w_i(\lambda-\theta_i)^{-1}=\lambda-\theta_0$$ for certain known weights $\{w_i\}$ and eigenvalues $\{\theta_i\}$ of a given matrix. This can be solved by truncating the sum to an increasing number of terms.
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PDF链接：
https://arxiv.org/pdf/802.0523