A unified approach to limit theorems for dual objects in probabilita and number theory

本项目的目的是扩展V. V. Buldygin，K. H.因德勒克岛I. Klesov和J. G. Steinebach关于“伪正则变函数和广义更新过程”（TBiMC，基辅，2012）研究各种对偶对象的极限定理的收敛速度。通过从分析真实的函数和序列的统一观点，我们的方法是导出概率论和数论中渐近陈述的几个等价性，并为其他领域发展类似的结果。例如，Marcinkiewicz-Zygmund型强大数定律或独立随机变量和的重对数定律与其相应的更新过程之间的关系是本项目主题中包含的概率论的具体结果。同样，极限分布和收敛速度的乘法和加法函数在数论及其扩展是一个进一步的例子的应用程序。该项目的另一个具有挑战性的问题是发现概率论中的数论结果的可能对应物，看起来非常类似于上述结果在更新theory. We一般的方法是基于伪正则变化函数的理论，利用其特定的属性，允许得出非常一般的渐近结论。这些类的功能的进一步发展也将进行调查，以扩大到广泛的应用领域的结果。

The aim of this project is to extend the general approach developed in the monograph by V. V. Buldygin, K.-H. Indlekofer, O. I. Klesov, and J. G. Steinebach on "Pseudo-Regularly Varying Functions and Generalized Renewal Processes" (TBiMC, Kyiv, 2012) to study the rate of convergence in limit theorems for various dual objects.Via a unified point of view from the analysis of real functions and sequences, our approach is to derive several equivalencies of asymptotic statements in probability theory as well as in number theory and to develop similar results for other fields of interest.For example, relationships between the Marcinkiewicz-Zygmund type strong laws of large numbers or laws of the iterated logarithm for sums of independent random variables and their corresponding renewal processes are specific results from probability theory included in the topics of this project.Similarly, limiting distributions and rates of convergence for multiplicative and additive functions in number theory and their extensions are a further example of applications. Another challenging problem of the project is to discover possible counterparts of number theoretical results in probability theory, looking very similar to the above mentioned results in renewal theory.Our general approach is based on the theory of pseudo-regularly varying functions, taking advantage of their specific properties which allow to draw very general asymptotic conclusions. The further development of such classes of functions will also be investigated in order to extend the results to a wide area of applications.