Summary
Consider a random walk S n on the integers, where the steps ξ i have mean 0 and variance σ2. Let T be the time of first self-intersection of the random walk. It is shown that, as σ→∞, T grows at rate σ2/3. More precisely, Tσ2/3 has a non-degenerate limit distribution which can be described in terms of Brownian motion local time.
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Research supported by National Science Foundation Grant MCS80-02698.
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Aldous, D.J. Self-intersections of 1-dimensional random walks. Probab. Th. Rel. Fields 72, 559–587 (1986). https://doi.org/10.1007/BF00344721
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DOI: https://doi.org/10.1007/BF00344721