随机可加位移下平稳点过程的信息容量和误差指数
Information-Theoretic Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements
摘要：
本文研究了平稳点过程随机位移的Shannon区域。设$\R^n$中某个初始驻点过程的每一点都产生一个子点，子点的位置由母点的坐标加上一个随机向量得到，所有的位移向量对所有点独立同分布.解码问题是这样的：已知整个母点过程以及某些子点的坐标;位移仅通过它们的定律已知;人们能找到这个子点的母点吗？香农机制是其中维度$n$趋于无穷大并且其中点过程的强度的对数与$n$成比例的机制。我们发现，这个问题表现出一个尖锐的阈值：如果噪声的比例因子和微分熵率的总和是积极的，那么找到正确的母点的概率趋于0与$n$的所有点过程和解码策略。如果该和为负，则存在母点过程（例如泊松）和解码策略（例如最大似然），对于该解码策略，找到正确的母节点的概率在$n$的情况下趋于1。然后，我们使用大偏差理论表明，在后一种情况下，如果噪声的熵谱满足大偏差原理，则错误概率以指数方式快速地变为0，其指数以噪声熵谱的速率函数的封闭形式给出。这是两类母点过程：泊松和Matérn。信息论的实际利益来自明确的连接，我们也建立了这个问题和香农的加性噪声信道的码字上的功率约束的误差指数的估计。
英文摘要：
This paper studies the Shannon regime for the random displacement of stationary point processes. Let each point of some initial stationary point process in $\R^n$ give rise to one daughter point, the location of which is obtained by adding a random vector to the coordinates of the mother point, with all displacement vectors independently and identically distributed for all points. The decoding problem is then the following one: the whole mother point process is known as well as the coordinates of some daughter point; the displacements are only known through their law; can one find the mother of this daughter point? The Shannon regime is that where the dimension $n$ tends to infinity and where the logarithm of the intensity of the point process is proportional to $n$. We show that this problem exhibits a sharp threshold: if the sum of the proportionality factor and of the differential entropy rate of the noise is positive, then the probability of finding the right mother point tends to 0 with $n$ for all point processes and decoding strategies. If this sum is negative, there exist mother point processes, for instance Poisson, and decoding strategies, for instance maximum likelihood, for which the probability of finding the right mother tends to 1 with $n$. We then use large deviations theory to show that in the latter case, if the entropy spectrum of the noise satisfies a large deviation principle, then the error probability goes exponentially fast to 0 with an exponent that is given in closed form in terms of the rate function of the noise entropy spectrum. This is done for two classes of mother point processes: Poisson and Matérn. The practical interest to information theory comes from the explicit connection that we also establish between this problem and the estimation of error exponents in Shannon's additive noise channel with power constraints on the codewords.
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