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Sylvester’s problem for random walks and bridges

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  • Panzo, Hugo

Abstract

Consider a random walk in Rd that starts at the origin and whose increment distribution assigns zero probability to any affine hyperplane. We solve Sylvester’s problem for these random walks by showing that the probability that any d+2 consecutive steps of the walk are in convex position is equal to 1−2(d+1)!. The analogous result also holds for random bridges of length at least d+2 whose joint increment distribution is exchangeable.

Suggested Citation

  • Panzo, Hugo, 2025. "Sylvester’s problem for random walks and bridges," Statistics & Probability Letters, Elsevier, vol. 219(C).
    • Handle: RePEc:eee:stapro:v:219:y:2025:i:c:s0167715224003183
    DOI: 10.1016/j.spl.2024.110349
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