LARGE DEVIATIONS OF THE EMPIRICAL VOLUME FRACTION FOR STATIONARY POISSON GRAIN MODELS
LARGE DEVIATIONS OF THE EMPIRICAL VOLUME FRACTION FOR STATIONARY POISSON GRAIN MODELS
复制标题
静止泊松粒模型经验体积分数的大偏差
DOI：
--
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
L. Heinrich
中科院分区：
文献类型：
--
作者：
L. Heinrich
我们研究了经验体积分数| {}\Xi\cap W＿n|/|W＿n|的尺度累积量生成函数L＿n(z)=|W＿n|^-1\logE\exp{z|\Xi\cap W_n|}（热力学）极限的存在性，其中| \cdot |表示d维勒贝格测度。这里\Xi = \bigcup ＿i{\ge 1}（\Xi ＿i+X＿i）表示一个由平稳泊松过程\Pi ＿ {\lambda} = \sum ＿i {\ge 1 }\delta ＿X＿i{定义的d维泊松颗粒模型（也称为布尔模型），其强度为}\lambda >0和一系列独立拷贝\Xi ＿1, \Xi ＿2，…一个随机紧集\Xi ＿0。对于{向各方向无界扩展W_n， n\ge 1，证明了当E }{}\exp{a|\Xi_0|}{ 0时在复平面上某盘上极限lim_n }{\to}{\infty} L＿n(z)的存在性和可解析性。此外，与此结果密切相关，我们得到了cram<s:1>和Chernoff意义上的经验体积分数的大偏差概率的指数不等式和精确渐近性。
We study the existence of the (thermodynamic) limit of the scaled cumulant-generating function L_n(z)=|W_n|^{-1}\logE\exp{z|\Xi\cap W_n|} of the empirical volume fraction |\Xi\cap W_n|/|W_n|, where |\cdot| denotes the d-dimensional Lebesgue measure. Here \Xi=\bigcup_{i\ge1}(\Xi_i+X_i) denotes a d-dimensional Poisson grain model (also known as a Boolean model) defined by a stationary Poisson process \Pi_{\lambda}=\sum_{i\ge1}\delta_{X_i} with intensity \lambda >0 and a sequence of independent copies \Xi_1,\Xi_2,... of a random compact set \Xi_0. For an increasing family of compact convex sets {W_n, n\ge1} which expand unboundedly in all directions, we prove the existence and analyticity of the limit lim_{n\to\infty}L_n(z) on some disk in the complex plane whenever E\exp{a|\Xi_0|} 0. Moreover, closely connected with this result, we obtain exponential inequalities and the exact asymptotics for the large deviation probabilities of the empirical volume fraction in the sense of Cram\'er and Chernoff.