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Spitzer Series and Regularly Varying Functions

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Pseudo-Regularly Varying Functions and Generalized Renewal Processes

Part of the book series: Probability Theory and Stochastic Modelling ((PTSM,volume 91))

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Abstract

Let X, {X n}n≥1 be independent, identically distributed random variables with distribution function F and let {S n}n≥1 be the sequence of their partial sums. Let w and φ be two positive functions. Put w k = w(k) and φ k = 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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Author information

Authors and Affiliations

  1. Department of Mathematical Analysis, National Technical University of Ukraine, Kyiv, Ukraine

    Valeriĭ V. Buldygin

  2. Department of Mathematics, University of Paderborn, Paderborn, Germany

    Karl-Heinz Indlekofer

  3. Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine, Kyiv, Ukraine

    Oleg I. Klesov

  4. Mathematical Institute, University of Cologne, Cologne, Germany

    Josef G. Steinebach

Authors
  1. Valeriĭ V. Buldygin
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  2. Karl-Heinz Indlekofer
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  3. Oleg I. Klesov
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  4. Josef G. Steinebach
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Buldygin, V.V., Indlekofer, KH., Klesov, O.I., Steinebach, J.G. (2018). Spitzer Series and Regularly Varying Functions. In: Pseudo-Regularly Varying Functions and Generalized Renewal Processes. Probability Theory and Stochastic Modelling, vol 91. Springer, Cham. https://doi.org/10.1007/978-3-319-99537-3_11

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