Invariance principles for random walks conditioned to stay positive

Caravenna, Francesco; Chaumont, Loïc

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 170-190.

Soit $\left\{S_n\right\}$ une marche aléatoire dont la loi est dans le domaine d’attraction d’une loi stable $𝒴$, i.e. il existe une suite de réels positifs $\left(a_n\right)$ telle que $S_n/a_n$ converge en loi vers $𝒴$. Nous montrons que le processus renormalisé $(S_{⌊nt⌋}/a_n,t\ge 0)$, une fois conditionné à rester positif, converge en loi (au sens fonctionnel) vers le processus de Lévy stable de loi $𝒴$ conditionné à rester positif. Sous certaines hypothèses supplémentaires, nous montrons un principe d’invariance pour cette marche aléatoire tuée lorsqu’elle quitte la demi-droite positive et conditionnée à mourir en 0.

Let $\left\{S_n\right\}$ be a random walk in the domain of attraction of a stable law $𝒴$, i.e. there exists a sequence of positive real numbers $\left(a_n\right)$ such that $S_n/a_n$ converges in law to $𝒴$. Our main result is that the rescaled process $(S_{⌊nt⌋}/a_n,t\ge 0)$, when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.

Classification : 60G18, 60G51, 60B10
Mots clés : random walk, stable law, Lévy process, conditioning to stay positive, invariance principle

L. Alili and R. A. Doney. Wiener-Hopf factorization revisited and some applications. Stoc. Stoc. Rep. 66 (1999) 87-102. | MR | Zbl

J. Bertoin. Lévy Processes. Cambridge University Press, 1996. | MR | Zbl

J. Bertoin and R. A. Doney. On conditioning a random walk to stay nonnegative. Ann. Probab. 22 (1994) 2152-2167. | MR | Zbl

N. H. Bingham, C. H. Goldie and J. L. Teugels. Regular Variation. Cambridge University Press, 1989. | MR | Zbl

P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1999. | MR | Zbl

E. Bolthausen. On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4 (1976) 480-485. | MR | Zbl

A. Bryn-Jones and R. A. Doney. A functional central limit theorem for random walks conditional to stay non-negative. J. London Math. Soc. (2) 74 (2006) 244-258. | MR | Zbl

F. Caravenna. A local limit theorem for random walks conditioned to stay positive. Probab. Theory Related Fields 133 (2005) 508-530. | MR | Zbl

F. Caravenna, G. Giacomin and L. Zambotti. Sharp asymptotic behavior for wetting models in (1+1)-dimension. Elect. J. Probab. 11 (2006) 345-362. | MR | Zbl

J.-D. Deuschel, G. Giacomin and L. Zambotti. Scaling limits of equilibrium wetting models in (1+1)-dimension. Probab. Theory Related Fields 132 (2005) 471-500. | MR | Zbl

L. Chaumont. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996) 39-54. | MR | Zbl

L. Chaumont. Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121 (1997) 377-403. | MR | Zbl

C. Dellacherie and P.-A. Meyer. Probabilités et potentiel. Chapitres XII-XVI. Théorie du potentiel associée à une résolvante. Théorie des processus de Markov, 2nd edition, 1417. Hermann, Paris, 1987. | MR | Zbl

R. A. Doney. Conditional limit theorems for asymptotically stable random walks. Z. Wahrsch. Verw. Gebiete 70 (1985) 351-360. | MR | Zbl

R. A. Doney and P. E. Greenwood. On the joint distribution of ladder variables of random walks. Probab. Theory Related Fields 94 (1993) 457-472. | MR | Zbl

R. A. Doney. Spitzer's condition and ladder variables in random walks. Probab. Theory Related Fields 101 (1995) 577-580. | MR | Zbl

R. A. Doney. One-sided local large deviation and renewal theorem in the case of infinite mean. Probab. Theory Related Fields 107 (1997) 451-465. | MR | Zbl

T. Duquesne and J. F. Le Gall. Lévy processes and spatial branching processes. Astérisque 281 (2002). | Numdam | MR | Zbl

S. N. Ethier and T. G. Kurtz. Markov Processes. Characterization and Convergence. Wiley, New York, 1986. | MR | Zbl

W. Feller. An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edition. Wiley, New York, 1971. | MR | Zbl

A. Garsia and J. Lamperti. A discrete renewal theorem with infinite mean. Comm. Math. Helv. 37 (1963) 221-234. | MR | Zbl

G. Giacomin. Random Polymer Models. Imperial College Press, World Scientific, 2007. | MR | Zbl

P. E. Greenwood, E. Omey and J. L. Teugels. Harmonic renewal measures and bivariate domains of attraction in fluctuation theory. Z. Wahrsch. Verw. Gebiete 61 (1982) 527-539. | MR | Zbl

D. L. Iglehart. Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2 (1974) 608-619. | MR | Zbl

A. Lambert. The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields 122 (2002) 42-70. | MR | Zbl

T. L. Liggett. An invariance principle for conditioned sums of independent random variables. J. Math. Mech. 18 (1968) 559-570. | MR | Zbl

A. V. Skorohod. Limit theorems for stochastic processes with independent increments. Theory Probab. Appl. 2 (1957) 138-171. | MR | Zbl

Y. Velenik. Localization and Delocalization of Random Interfaces. Probab. Surv. 3 (2006) 112-169. | MR