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On the centre of mass of a random walk

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  • Lo, Chak Hei
  • Wade, Andrew R.

Abstract

For a random walk Sn on Rd we study the asymptotic behaviour of the associated centre of mass process Gn=n−1∑i=1nSi. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, Gn is recurrent if d=1 and transient if d≥2. In the transient case we show that Gn has a diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which Gn is transient in d=1.

Suggested Citation

  • Lo, Chak Hei & Wade, Andrew R., 2019. "On the centre of mass of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4663-4686.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:11:p:4663-4686
    DOI: 10.1016/j.spa.2018.12.007
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    References listed on IDEAS

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    1. Grill, Karl, 1988. "On the average of a random walk," Statistics & Probability Letters, Elsevier, vol. 6(5), pages 357-361, April.
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    Cited by:

    1. Bercu, Bernard & Laulin, Lucile, 2021. "On the center of mass of the elephant random walk," Stochastic Processes and their Applications, Elsevier, vol. 133(C), pages 111-128.

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