论文标题
古典且几乎确定的本地限制定理
Classical and Almost Sure Local Limit Theorems
论文作者
论文摘要
我们介绍并讨论了有关著名限制定理的许多结果，即局部极限定理，该定理具有许多接口，具有数字理论，尽管如此，尽管如此，尽管如此，尽管如此，但对于局部限制定理的有效性条件的问题，至今却没有令人满意的解决方案。这些结果主要涉及局部极限定理的有效性及其有趣的变体形式的足够条件：强局限制定理，强大的局部极限定理具有变化的收敛。非常重要的是必要的条件，而获得的结果本质上是稀疏的：Rozanov的必要条件，Gamkrelidze的必要条件，以及Mukhin的必要条件。由于Azlarov和Gamkrelidze引起的反例，以及对一类随机变量获得的必要条件，例如Mitalauskas对具有稳定极限分布的随机变量的强大形式的表征。提出并比较了特征函数和Bernoulli部分提取方法的方法。该调查的第二部分致力于对Denker和Koch灌输的几乎确定的局部限制定理的最新研究。建立的几乎确定的本地限制定理已经涵盖I.I.D.案例，稳定的情况，马尔可夫链，迪克曼功能的模型。我们在撰写本专着的目的是让知识渊博于六十年代及之后的立陶宛和俄罗斯流派获得的许多有趣的结果，这些结果基本上是用俄语写的，而且经常在困难获得期刊上发表。
We present and discuss the many results obtained concerning a famous limit theorem, the local limit theorem, which has many interfaces, with Number Theory notably, and for which, in spite of considerable efforts, the question concerning conditions of validity of the local limit theorem, has up to now no satisfactory solution. These results mostly concern sufficient conditions for the validity of the local limit theorem and its interesting variant forms: strong local limit theorem, strong local limit theorem with convergence in variation. Quite importantly are necessary conditions, and the results obtained are sparse, essentially: Rozanov's necessary condition, Gamkrelidze's necessary condition, and Mukhin's necessary and sufficient condition. Extremely useful and instructive are the counter-examples due to Azlarov and Gamkrelidze, as well as necessary and sufficient conditions obtained for a class of random variables, such as Mitalauskas' characterization of the local limit theorem in the strong form for random variables having stable limit distributions. The method of characteristic functions and the Bernoulli part extraction method, are presented and compared. A second part of the survey is devoted to the more recent study of the almost sure local limit theorem, instilled by Denker and Koch. The almost sure local limit theorems established already cover the i.i.d. case, the stable case, Markov chains, the model of the Dickman function. Our aim in writing this monograph was notably to bring to knowledge many interesting results obtained by the Lithuanian and Russian Schools of Probability during the sixties and after, and which are essentially written in Russian, and moreover often published in Journals of difficult access.
