Points of increase for random walks |
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| Authors: | Yuval Peres |
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| Institution: | (1) Department of Statistics, University of California, 367 Evans Hall, 94720 Berkeley, CA, USA;(2) Present address: Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel |
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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.
Research partially supported by NSF grant # DMS-9404391. |
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