Geodesic
Convex Minorants of Random Walks and Brownian Motion
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 3, pp. 498-512

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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, Brownian motion
Mots-clés : convex minorant, random permutations. 
@article{TVP_2001_46_3_a5,
     author = {T. M. Suidan},
     title = {Convex {Minorants} of {Random} {Walks} and {Brownian} {Motion}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {498--512},
     publisher = {mathdoc},
     volume = {46},
     number = {3},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2001_46_3_a5/}
}
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T. M. Suidan. Convex Minorants of Random Walks and Brownian Motion. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 3, pp. 498-512. http://geodesic.mathdoc.fr/item/TVP_2001_46_3_a5/