An invariance principle for random walk bridges conditioned to stay positive

Francesco Caravenna (Università degli Studi di Milano-Bicocca)
Loïc Chaumont (Université d'Angers)

Abstract


We prove an invariance principle for the bridge of a random walk conditioned to stay positive, when the random walk is in the domain of attraction of a stable law, both in the discrete and in the absolutely continuous setting. This includes as a special case the convergence under diffusive rescaling of random walk excursions toward the normalized Brownian excursion, for zero mean, finite variance random walks. The proof exploits asuitable absolute continuity relation together with some local asymptotic estimates for random walks conditioned to stay positive, recently obtained by Vatutin and Wachtel and by Doney.We review and extend these relations to the absolutely continuous setting.

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Pages: 1-32

Publication Date: June 5, 2013

DOI: 10.1214/EJP.v18-2362

References

  • Alili, Larbi; Chaumont, Loïc. A new fluctuation identity for Lévy processes and some applications. Bernoulli 7 (2001), no. 3, 557-569. MR1836746
  • Alili, L.; Doney, R. A. Wiener-Hopf factorization revisited and some applications. Stochastics Stochastics Rep. 66 (1999), no. 1-2, 87-102. MR1687803
  • Asmussen, Søren; Foss, Serguei; Korshunov, Dmitry. Asymptotics for sums of random variables with local subexponential behaviour. J. Theoret. Probab. 16 (2003), no. 2, 489-518. MR1982040
  • Bertoin, Jean. Lévy processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge, 1996. x+265 pp. ISBN: 0-521-56243-0 MR1406564
  • Bertoin, Jean. Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl. 47 (1993), no. 1, 17-35. MR1232850
  • Bertoin, J.; Doney, R. A. On conditioning a random walk to stay nonnegative. Ann. Probab. 22 (1994), no. 4, 2152-2167. MR1331218
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9 MR1700749
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1989. xx+494 pp. ISBN: 0-521-37943-1 MR1015093
  • Bolthausen, Erwin. On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probability 4 (1976), no. 3, 480-485. MR0415702
  • Bryn-Jones, A.; Doney, R. A. A functional limit theorem for random walk conditioned to stay non-negative. J. London Math. Soc. (2) 74 (2006), no. 1, 244-258. MR2254563
  • Caravenna, Francesco. A local limit theorem for random walks conditioned to stay positive. Probab. Theory Related Fields 133 (2005), no. 4, 508-530. MR2197112
  • F. Caravenna, A note on directly Riemann integrable functions, arXiv:1210.2361.
  • Caravenna, Francesco; Chaumont, Loïc. Invariance principles for random walks conditioned to stay positive. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 1, 170-190. MR2451576
  • Caravenna, Francesco; Giacomin, Giambattista; Zambotti, Lorenzo. A renewal theory approach to periodic copolymers with adsorption. Ann. Appl. Probab. 17 (2007), no. 4, 1362-1398. MR2344310
  • Caravenna, Francesco; Giacomin, Giambattista; Zambotti, Lorenzo. Sharp asymptotic behavior for wetting models in $(1+1)$-dimension. Electron. J. Probab. 11 (2006), no. 14, 345-362 (electronic). MR2217821
  • Chaumont, L. Excursion normalisée, méandre et pont pour les processus de Lévy stables. (French) [Normalized excursion, meander and bridge for stable Levy processes] Bull. Sci. Math. 121 (1997), no. 5, 377-403. MR1465814
  • Chaumont, L.; Doney, R. A. On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 (2005), no. 28, 948-961. MR2164035
  • Chaumont, L.; Doney, R. A. Invariance principles for local times at the maximum of random walks and Lévy processes. Ann. Probab. 38 (2010), no. 4, 1368-1389. MR2663630
  • Doney, R. A. Conditional limit theorems for asymptotically stable random walks. Z. Wahrsch. Verw. Gebiete 70 (1985), no. 3, 351-360. MR0803677
  • Doney, R. A.; Greenwood, P. E. On the joint distribution of ladder variables of random walk. Probab. Theory Related Fields 94 (1993), no. 4, 457-472. MR1201554
  • Doney, R. A. Local behaviour of first passage probabilities. Probab. Theory Related Fields 152 (2012), no. 3-4, 559-588. MR2892956
  • Doney, R. A.; Savov, M. S. The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab. 38 (2010), no. 1, 316-326. MR2599201
  • Donsker, Monroe D. An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc., 1951, (1951). no. 6, 12 pp. MR0040613
  • Èppelʹ, M. S. A local limit theorem for the first passage time. (Russian) Sibirsk. Mat. Zh. 20 (1979), no. 1, 181-191, 207. MR0523146
  • W. Feller, textitAn Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed. (1971), John Wiley and Sons, New York. MR0270403 MR0270403
  • Fitzsimmons, Pat; Pitman, Jim; Yor, Marc. Markovian bridges: construction, Palm interpretation, and splicing. Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), 101--134, Progr. Probab., 33, Birkhäuser Boston, Boston, MA, 1993. MR1278079
  • Giacomin, Giambattista. Random polymer models. Imperial College Press, London, 2007. xvi+242 pp. ISBN: 978-1-86094-786-5; 1-86094-786-7 MR2380992
  • Giacomin, Giambattista. Disorder and critical phenomena through basic probability models. Lecture notes from the 40th Probability Summer School held in Saint-Flour, 2010. Lecture Notes in Mathematics, 2025. École d'Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School] Springer, Heidelberg, 2011. xii+130 pp. ISBN: 978-3-642-21155-3 MR2816225
  • P. E. Greenwood, E. Omey and J. L. Teugels, textitHarmonic renewal measures and bivariate domains of attraction in fluctuation theory, J. Aust. Math. Soc., Ser. A 32 (1982), 412-422. MR0682578
  • Gnedenko, B. V.; Kolmogorov, A. N. Limit distributions for sums of independent random variables. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954. ix+264 pp. MR0062975
  • den Hollander, Frank. Random polymers. Lectures from the 37th Probability Summer School held in Saint-Flour, 2007. Lecture Notes in Mathematics, 1974. Springer-Verlag, Berlin, 2009. xiv+258 pp. ISBN: 978-3-642-00332-5 MR2504175
  • Iglehart, Donald L. Functional central limit theorems for random walks conditioned to stay positive. Ann. Probability 2 (1974), 608-619. MR0362499
  • Kaigh, W. D. An invariance principle for random walk conditioned by a late return to zero. Ann. Probability 4 (1976), no. 1, 115-121. MR0415706
  • Kozlov, M. V. The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment. (Russian) Teor. Verojatnost. i Primenen. 21 (1976), no. 4, 813-825. MR0428492
  • Liggett, Thomas M. An invariance principle for conditioned sums of independent random variables. J. Math. Mech. 18 1968 559-570. MR0238373
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • B. A. Rogozin, textitThe Distribution of the First Ladder Moment and Height and Fluctuation of a Random Walk, Theory Probab. Appl. 21 (1976), 575-595. MR0290473
  • Ya. G. Sinai, textitOn the Distribution of the First Positive Sum for a Sequence of Independent Random Variables, Theory Probab. Appl. 2 (1957), 122-129.
  • J. Sohier, textitA functional limit convergence towards brownian excursion, arXiv:1012.0118.
  • Skorohod, A. V. Limit theorems for stochastic processes with independent increments. (Russian) Teor. Veroyatnost. i Primenen. 2 1957 145--177. MR0094842
  • G. Uribe Bravo, textitBridges of Lévy processes conditioned to stay positive, Bernoulli (to appear), arXiv:1101.4184.
  • Vatutin, Vladimir A.; Wachtel, Vitali. Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 (2009), no. 1-2, 177-217. MR2449127
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