Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Exit times for integrated random walks

Denis Denisov and Vitali Wachtel

Full-text: Open access


We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time $n$. Assuming that the moment of order $2+\delta$ is finite, we show that the exact asymptotics for this probability is $n^{-1/4}$. To show this asymptotics we develop a discrete potential theory for integrated random walks.


Nous considérons une marche aléatoire centrée de variance finie et étudions le comportement asymptotique de la probabilité que l’aire sous la marche reste positive jusqu’à un grand temps $n$. Si le moment d’ordre $2+\delta$ est fini, nous montrons que cette probabilité décroit comme $n^{-1/4}$. Pour prouver ce comportement asymptotique, nous développons une théorie du potentiel discrète pour des marches aléatoires intégrées.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 1 (2015), 167-193.

First available in Project Euclid: 14 January 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60F17: Functional limit theorems; invariance principles

Markov chain Exit time Harmonic function Weyl chamber Normal approximation Kolmogorov diffusion


Denisov, Denis; Wachtel, Vitali. Exit times for integrated random walks. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 1, 167--193. doi:10.1214/13-AIHP577.

Export citation


  • [1] M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions Formulas, Graphs and Mathematical Tables. Dover Publications Inc., New York. 1992. Reprint of the 1972 edition.
  • [2] F. Aurzada and S. Dereich. Universality of asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 236–251.
  • [3] F. Aurzada, S. Dereich and M. Lifshits. Persistence probabilities for an integrated random walk bridge. Available at arXiv:1205.2895, 2012.
  • [4] F. Caravenna and J.-D. Deuschel. Pinning and wetting transition in $(1+1)$-dimensional fields with Laplacian interaction. Ann. Probab. 36 (2008) 2388–2433.
  • [5] A. Dembo, J. Ding and F. Gao. Persistence of iterated partial sums. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 873–884. Available at arXiv:1205.5596.
  • [6] D. Denisov and V. Wachtel. Conditional limit theorems for ordered random walks. Electron. J. Probab. 15 (2010) 292–322.
  • [7] D. Denisov and V. Wachtel. Random walks in cones. Ann. Probab. To appear. Available at arXiv:1110.1254, 2011.
  • [8] O. Friedland and S. Sodin. Bounds on the concentration function in terms of the Diophantine approximation. C. R. Math. Acad. Sci. Paris 345 (2007) 513–518.
  • [9] P. Groeneboom, G. Jongbloed and J. A. Wellner. Integrated Brownian motion, conditioned to be positive. Ann. Probab. 27 (1999) 1283–1303.
  • [10] A. Lachal. Sur les excursions de l’intégrale du mouvement brownien. C. R. Acad. Sci. Paris Sér. I Math. 314 (1992) 1053–1056.
  • [11] P. Major. The approximation of partial sums of RV’s. Z. Wahrsch. verw. Gebiete 35 (1976) 213–220.
  • [12] H. P. McKean. Jr. A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227–235.
  • [13] Ya. G. Sinai. Distribution of some functionals of the integral of a random walk. Theor. Math. Phys. 90 (1992) 219–241.
  • [14] V. Vysotsky. On the probability that integrated random walks stay positive. Stochastic Process. Appl. 120 (2010) 1178–1193.
  • [15] V. Vysotsky. Positivity of integrated random walks. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014) 195–213.