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Limit theorems for random walks

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  • Bendikov, Alexander
  • Cygan, Wojciech
  • Trojan, Bartosz

Abstract

We consider a random walk Sτ which is obtained from the simple random walk S by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator τ appropriately scaled random walk Sτ converges in the Skorohod space to the symmetric α-stable process Bα. We also prove asymptotic formula for the transition function of Sτ similar to the Pólya’s asymptotic formula for Bα.

Suggested Citation

  • Bendikov, Alexander & Cygan, Wojciech & Trojan, Bartosz, 2017. "Limit theorems for random walks," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3268-3290.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:10:p:3268-3290
    DOI: 10.1016/j.spa.2017.02.008
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    References listed on IDEAS

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    1. Ward Whitt, 1980. "Some Useful Functions for Functional Limit Theorems," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 67-85, February.
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    Cited by:

    1. Chang-Song Deng, 2020. "Subgeometric Rates of Convergence for Discrete-Time Markov Chains Under Discrete-Time Subordination," Journal of Theoretical Probability, Springer, vol. 33(1), pages 522-532, March.
    2. Wojciech Cygan & Stjepan Šebek, 2021. "Transition probability estimates for subordinate random walks," Mathematische Nachrichten, Wiley Blackwell, vol. 294(3), pages 518-558, March.

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