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On the average of a random walk

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  • Grill, Karl

Abstract

Let {Sn, n [epsilon] N)} be a simple random walk and denote by An its time average: An = (S1+ ...+Sn)/n. We give an integral test for the lower bound on An, thus giving an affirmative answer to a conjecture of P. Erdös (private communication) that An will return to a fixed region around the origin infinitely often with probability 1 in 1 dimension whereas in 2 or more dimensions it will return only finitely many times.

Suggested Citation

  • Grill, Karl, 1988. "On the average of a random walk," Statistics & Probability Letters, Elsevier, vol. 6(5), pages 357-361, April.
    • Handle: RePEc:eee:stapro:v:6:y:1988:i:5:p:357-361
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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.
    2. Hryniv, Ostap & Menshikov, Mikhail V. & Wade, Andrew R., 2013. "Excursions and path functionals for stochastic processes with asymptotically zero drifts," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1891-1921.
    3. 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.

    More about this item

    Keywords

    random walk recurrence strong laws;

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