A central limit theorem for two-dimensional random walks in a cone

Rodolphe Garbit

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 2, page 271-286
  • ISSN: 0037-9484

Abstract

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We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying. This condition is satisfied in many natural examples.

How to cite

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Garbit, Rodolphe. "A central limit theorem for two-dimensional random walks in a cone." Bulletin de la Société Mathématique de France 139.2 (2011): 271-286. <http://eudml.org/doc/272623>.

@article{Garbit2011,
abstract = {We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying. This condition is satisfied in many natural examples.},
author = {Garbit, Rodolphe},
journal = {Bulletin de la Société Mathématique de France},
keywords = {conditioned random walks; brownian motion; brownian meander; cone; functional limit theorem; regularly varying sequences},
language = {eng},
number = {2},
pages = {271-286},
publisher = {Société mathématique de France},
title = {A central limit theorem for two-dimensional random walks in a cone},
url = {http://eudml.org/doc/272623},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Garbit, Rodolphe
TI - A central limit theorem for two-dimensional random walks in a cone
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 2
SP - 271
EP - 286
AB - We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying. This condition is satisfied in many natural examples.
LA - eng
KW - conditioned random walks; brownian motion; brownian meander; cone; functional limit theorem; regularly varying sequences
UR - http://eudml.org/doc/272623
ER -

References

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  3. [3] E. Bolthausen – « On a functional central limit theorem for random walks conditioned to stay positive », Ann. Probability4 (1976), p. 480–485. Zbl0336.60024MR415702
  4. [4] R. T. Durrett, D. L. Iglehart & D. R. Miller – « Weak convergence to Brownian meander and Brownian excursion », Ann. Probability5 (1977), p. 117–129. Zbl0356.60034MR436353
  5. [5] R. Garbit – « Contributions à l’étude d’une marche aléatoire centrifuge et théorèmes limites pour des processus aléatoires conditionnés », Thèse, Université de Tours, 2008. 
  6. [6] —, « Brownian motion conditioned to stay in a cone », J. Math. Kyoto Univ.49 (2009), p. 573–592. Zbl1192.60091MR2583602
  7. [7] D. L. Iglehart – « Functional central limit theorems for random walks conditioned to stay positive », Ann. Probability2 (1974), p. 608–619. Zbl0299.60053MR362499
  8. [8] R. Lang – « A note on the measurability of convex sets », Arch. Math. (Basel) 47 (1986), p. 90–92. Zbl0607.28003MR855142
  9. [9] M. Shimura – « Excursions in a cone for two-dimensional Brownian motion », J. Math. Kyoto Univ.25 (1985), p. 433–443. Zbl0582.60048MR807490
  10. [10] —, « A limit theorem for two-dimensional random walk conditioned to stay in a cone », Yokohama Math. J.39 (1991), p. 21–36. Zbl0741.60068MR1137264
  11. [11] F. Spitzer – « A Tauberian theorem and its probability interpretation », Trans. Amer. Math. Soc.94 (1960), p. 150–169. Zbl0216.21201MR111066
  12. [12] N. T. Varopoulos – « Potential theory in conical domains », Math. Proc. Cambridge Philos. Soc.125 (1999), p. 335–384. Zbl0918.31008MR1643806
  13. [13] I. Weissman – « A note on Bojanic-Seneta theory of regularly varying sequences », Math. Z.151 (1976), p. 29–30. Zbl0319.40003MR417349

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