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Points of increase for random walks

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Abstract

Say that a sequenceS 0, ..., Sn has a (global) point of increase atk ifS k is maximal amongS 0, ..., Sk and minimal amongS k, ..., Sn. We give an elementary proof that ann-step symmetric random walk on the line has a (global) point of increase with probability comparable to 1/logn. (No moment assumptions are needed.) This implies the classical fact, due to Dvoretzky, Erdős and Kakutani (1961), that Brownian motion has no points of increase.

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Author notes

  1. Yuval Peres

    Present address: Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

Authors and Affiliations

  1. Department of Statistics, University of California, 367 Evans Hall, 94720, Berkeley, CA, USA

    Yuval Peres

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  1. Yuval Peres
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Correspondence to Yuval Peres.

Additional information

Research partially supported by NSF grant # DMS-9404391.

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Peres, Y. Points of increase for random walks. Israel J. Math. 95, 341–347 (1996). https://doi.org/10.1007/BF02761045

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  • DOI: https://doi.org/10.1007/BF02761045

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