Equivalence of Limit Theorems for Sums of Random Variables and Renewal Processes

  • Valeriĭ V. Buldygin
  • Karl-Heinz Indlekofer
  • Oleg I. Klesov
  • Josef G. Steinebach
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 91)


Many limit results are known for cumulative sums of independent identically distributed random variables and the corresponding renewal counting processes to hold under the same conditions. Despite the coincidence of conditions for both cases, the theories have been developed independently of each other and the methods are different. A general question is whether or not the whole theories of limit theorems for sums and renewal processes are equivalent. In this chapter, we consider the problem of the equivalence of certain asymptotic results, like the strong law of large numbers or the law of the iterated logarithm, for a sequence of sums of independent, identically distributed random variables and its corresponding renewal process.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Valeriĭ V. Buldygin
    • 1
  • Karl-Heinz Indlekofer
    • 2
  • Oleg I. Klesov
    • 3
  • Josef G. Steinebach
    • 4
  1. 1.Department of Mathematical AnalysisNational Technical University of UkraineKyivUkraine
  2. 2.Department of MathematicsUniversity of PaderbornPaderbornGermany
  3. 3.Department of Mathematical Analysis and Probability TheoryNational Technical University of UkraineKyivUkraine
  4. 4.Mathematical InstituteUniversity of CologneCologneGermany

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