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This article is cited in 7 scientific papers (total in 7 papers)
Convex Minorants of Random Walks and Brownian Motion
T. M. Suidan Princeton University
Abstract:
Let $(S_{i})_{i=0}^n$ be the random walk process generated by a sequence of real-valued independent identically distributed random variables $(X_{i})_{i=1}^n$ having densities. We study probability distributions related to the associated convex minorant process. In particular, we investigate the length of a convex minorant's longest segment. Using random permutation theory, we fully characterize the probability distribution of the length of the $r$th longest segment of the convex minorant generated by Brownian motion on finite intervals; we also give an explicit density for the joint distributions of the first $r$ longest segments. In addition, we use the methods developed here to prove Sparre Andersen's formula for the probability of having $m$ segments composing the convex minorant of a random walk of length $N$. We describe analogous statements for random walks with random time increments. The author has recently used these results to solve a problem of adhesion dynamics on the line.
Keywords:
random walk, convex minorant, Brownian motion, random permutations.
Received: 12.02.2001
Citation:
T. M. Suidan, “Convex Minorants of Random Walks and Brownian Motion”, Teor. Veroyatnost. i Primenen., 46:3 (2001), 498–512; Theory Probab. Appl., 46:3 (2002), 469–481
Linking options:
https://www.mathnet.ru/eng/tvp3898https://doi.org/10.4213/tvp3898 https://www.mathnet.ru/eng/tvp/v46/i3/p498
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