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Spitzer Series and Regularly Varying Functions

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

Abstract

Let X, {Xn}n≥1 be independent, identically distributed random variables with distribution function F and let {Sn}n≥1 be the sequence of their partial sums. Let w and φ be two positive functions. Put wk = w(k) and φk = φ(k). We study the convergence and the asymptotic behavior with respect to small parameters ε of the series
$$\displaystyle Q(\varepsilon )=\sum _{k=1}^\infty w_k P(|S_k| \ge \varepsilon \varphi _k),\qquad \varepsilon >0.$$
Main result of this chapter is Theorem 11.1, together with some corollaries, which exhibit possible asymptotics for simple choices of the functions w(⋅) and φ(⋅) in (11.1).

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Copyright information

© 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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