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

这个项目的目的是扩展在专着发展的一般方法V. V. Buldygin, k - h。Indlekofer， O. I. Klesov和J. G. Steinebach在“伪正则变函数和广义更新过程”（timc，基辅，2012）上研究了各种对偶对象的极限定理的收敛速度。通过实函数和实序列分析的统一观点，我们的方法是推导出概率论和数论中渐近陈述的几个等价，并为其他感兴趣的领域开发类似的结果。例如，Marcinkiewicz-Zygmund型强大数定律或独立随机变量和的迭代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，数论中乘法和加性函数的极限分布和收敛率及其扩展是应用的另一个例子。项目的另一个挑战问题是发现概率论中数论结果的可能对应物，看起来与上述更新理论中的结果非常相似。我们的一般方法是基于伪正则变函数理论，利用它们的特殊性质，可以得出非常一般的渐近结论。还将研究这类函数的进一步发展，以便将结果扩展到广泛的应用领域。

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.