Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-27T23:52:37.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Concave Majorants of Random Walks and Related Poisson Processes

Published online by Cambridge University Press:  18 August 2011

JOSH ABRAMSON  and
JIM PITMAN
JOSH ABRAMSON
Affiliation:
Department of Statistics, University of California at Berkeley, CA 94720, USA (e-mail: [email protected], [email protected])
JIM PITMAN
Affiliation:
Department of Statistics, University of California at Berkeley, CA 94720, USA (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We offer a unified approach to the theory of concave majorants of random walks, by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant of a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested compositions that naturally arise from our construction of the concave majorant.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011 , pp. 651 - 682
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bertoin, J. (1999) Renewal theory for embedded regenerative sets. Ann. Probab. 27 1523–1535.CrossRefGoogle Scholar
[2]Bertoin, J. (2000) The convex minorant of the Cauchy process. Electron. Comm. Probab. 5 51–55.CrossRefGoogle Scholar
[3]Brunk, H. D. (1964) A generalization of Spitzer's combinatorial lemma. Z. Wahrsch. Verw. Gebiete 2 395–405.CrossRefGoogle Scholar
[4]Diaconis, P., Holmes, S., Janson, S., Lalley, S. P. and Pemantle, R. (1996) Metrics on compositions and coincidences among renewal sequences. In Random Discrete Structures, Vol. 76 of IMA Volumes in Mathematics and its Applications, Springer, pp. 81–101.CrossRefGoogle Scholar
[5]Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II, second edition, Wiley.Google Scholar
[6]Gnedin, A. and Pitman, J. (2005) Regenerative composition structures. Ann. Probab. 33 445–479.CrossRefGoogle Scholar
[7]Goldie, C. M. (1989) Records, permutations and greatest convex minorants. Math. Proc. Cambridge Philos. Soc. 106 169–177.CrossRefGoogle Scholar
[8]Greenwood, P. and Pitman, J. (1980) Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Probab. 12 893–902.CrossRefGoogle Scholar
[9]Greenwood, P. and Pitman, J. (1980) Fluctuation identities for random walk by path decomposition at the maximum. Adv. Appl. Probab. 12 291–293.CrossRefGoogle Scholar
[10]Groeneboom, P. (1983) The concave majorant of Brownian motion. Ann. Probab. 11 1016–1027.CrossRefGoogle Scholar
[11]Kac, M. (1954) Toeplitz matrices, translation kernels and a related problem in probability theory. Duke Math. J. 21 501–509.CrossRefGoogle Scholar
[12]Pitman, J. (2006) Combinatorial Stochastic Processes, Vol. 1875 of Lecture Notes in Mathematics, Springer.Google Scholar
[13]Pitman, J. and Uribe Bravo, G. (2011) The convex minorant of a Lévy process. Ann. Probab., to appear.CrossRefGoogle Scholar
[14]Qiao, Z. and Steele, J. M. (2005) Random walks whose concave majorants often have few faces. Statist. Probab. Lett. 75 97–102.CrossRefGoogle Scholar
[15]Shepp, L. A. and Lloyd, S. P. (1966) Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 340–357.CrossRefGoogle Scholar
[16]Sherman, S. (1964) Fluctuation and periodicity. J. Math. Anal. Appl. 9 468–476.CrossRefGoogle Scholar
[17]Sparre Andersen, E. (1954) On the fluctuations of sums of random variables II. Math. Scand. 2 195–223.Google Scholar
[18]Sparre Andersen, E. (1959) On the distribution of the random variable H n. Tech. Sci. Note no. 1, Contract no. AF 61(052)-42.Google Scholar
[19]Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 323–339.CrossRefGoogle Scholar
[20]Steele, J. M. (2002) The Bohnenblust–Spitzer algorithm and its applications. J. Comput. Appl. Math. 142 235–249.CrossRefGoogle Scholar
[21]Vervaat, W. (1979) A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 143–149.CrossRefGoogle Scholar