Positivity of integrated random walks

Vysotsky, Vladislav

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 195-213 / Harvested from Numdam

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Abstract

Soit $S_{n}$ une marche aléatoire centrée, nous considérons la suite de ses sommes partielles $A_{n} : = \sum_{i = 1}^{n} S_{i}$ . Nous supposons que $S_{1}$ est dans le domaine d’attraction normale d’une loi $α$ -stable avec $1 < α \leq 2$ . En supposant que $S_{1}$ est soit exponentielle à droite (i.e. $ℙ (S_{1} > x | S_{1} > 0) = e^{- a x}$ ), soit continue à droite (i.e. $ℙ (S_{1} = 1 | S_{1} > 0) = 1$ ), nous prouvons que $ℙ {A_{1} > 0, \dots, A_{N} > 0} \sim C_{α} N^{1 / (2 α) - 1 / 2}$ quand $N \to \infty$ , où $C_{α} > 0$ dépend de la distribution de la marche. Nous considérons aussi une version conditionnelle de ce problème et nous étudions la positivité de ponts discrets intégrés.

Take a centered random walk $S_{n}$ and consider the sequence of its partial sums $A_{n} : = \sum_{i = 1}^{n} S_{i}$ . Suppose $S_{1}$ is in the domain of normal attraction of an $α$ -stable law with $1 < α \leq 2$ . Assuming that $S_{1}$ is either right-exponential (i.e. $ℙ (S_{1} > x | S_{1} > 0) = e^{- a x}$ for some $a > 0$ and all $x > 0$ ) or right-continuous (skip free), we prove that $ℙ {A_{1} > 0, \dots, A_{N} > 0} \sim C_{α} N^{1 / (2 α) - 1 / 2}$ as $N \to \infty$ , where $C_{α} > 0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Publié le : 2014-01-01

MR 3161528 | Zbl 1293.60053 | 1 citation dans Numdam

DOI : https://doi.org/10.1214/12-AIHP487
Classification: 60G50, 60F99

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     author = {Vysotsky, Vladislav},
     title = {Positivity of integrated random walks},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {50},
     year = {2014},
     pages = {195-213},
     doi = {10.1214/12-AIHP487},
     mrnumber = {3161528},
     zbl = {1293.60053},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2014__50_1_195_0}
}

Vysotsky, Vladislav. Positivity of integrated random walks. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) pp. 195-213. doi : 10.1214/12-AIHP487. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_1_195_0/

Bibliographie

[1] F. Aurzada and S. Dereich. Universality of the asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. Henri Poincaré Probab. Stat. 49(1) (2013) 236-251. | Numdam | MR 3060155 | Zbl 1285.60042

[2] J. Bertoin. Lévy Processes. Cambridge University Press, New York, 1996. | MR 1406564 | Zbl 0938.60005

[3] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1999. | MR 1700749 | Zbl 0172.21201

[4] E. Bolthausen. On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4 (1976) 480-485. | MR 415702 | Zbl 0336.60024

[5] F. Caravenna and J.-D. Deuschel. Pinning and wetting transition for ( $1 + 1$ )-dimensional fields with Laplacian interaction. Ann. Probab. 36 (2008) 2388-2433. | MR 2478687 | Zbl 1179.60066

[6] A. Dembo, J. Ding and F. Gao. Persistence of iterated partial sums. Ann. Inst. Henri Poincaré Probab. Stat. 49(3) (2013) 873-884. | Numdam | MR 3112437 | Zbl 1274.60144

[7] R. A. Doney. Conditional limit theorems for asymptotically stable random walks. Z. Wahrsch. Verw. Gebiete 70 (1985) 351-360. | MR 803677 | Zbl 0573.60063

[8] R. A. Doney. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 (1997) 451-465. | MR 1440141 | Zbl 0883.60022

[9] V. A. Egorov. On the rate of convergence to a stable law. Theor. Probab. Appl. 25 (1980) 180-187. | MR 560073 | Zbl 0456.60020

[10] W. Feller. An Introduction to Probability Theory and Its Applications 2. Wiley, New York, 1966. | MR 210154 | Zbl 0219.60003

[11] P. Greenwood and M. Shaked. Fluctuations of random walk in $R^{d}$ and storage systems. Adv. in Appl. Prob. 9 (1977) 566-587. | MR 464406 | Zbl 0376.60073

[12] I. A. Ibragimov and Yu. V. Linnik. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen, 1971. | MR 322926 | Zbl 0219.60027

[13] Y. Isozaki and S. Watanabe. An asymptotic formula for the Kolmogorov Diffusion and a refinement of Sinai's estimates for the integral of Brownian motion. Proc. Japan Acad. Ser. A 70 (1994) 271-276. | MR 1313176 | Zbl 0820.60066

[14] S. Janson. Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas. Probab. Surveys 4 (2007) 80-145. | MR 2318402 | Zbl 1189.60147

[15] H. Kesten. Ratio theorems for random walks II. J. Anal. Math. 11 (1963) 323-379. | MR 163365 | Zbl 0121.35202

[16] S. M. Majumdar. Persistence in nonequilibrium systems. Current Sci. 77 (1999) 370-375.

[17] S. V. Nagaev. On the asymptotic behaviour of one-sided large deviation probabilities. Theory Probab. Appl. 26 (1982) 362-366. | Zbl 0481.60036

[18] S. Resnick and P. Greenwood. A bivariate stable characterization and domains of attraction. J. Multivariate Anal. 9 (1979) 206-221. | MR 538402 | Zbl 0409.62038

[19] M. Shimura. A class of conditional limit theorems related to ruin problem. Ann. Probab. 11 (1983) 40-45. | MR 682799 | Zbl 0506.60066

[20] Ya. G. Sinai. Distribution of some functionals of the integral of a random walk. Theor. Math. Phys. 90 (1992) 219-241. | MR 1182301 | Zbl 0810.60063

[21] V. A. Vatutin and V. Wachtel. Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 (2009) 177-217. | MR 2449127 | Zbl 1158.60014

[22] V. Vysotsky. Clustering in a stochastic model of one-dimensional gas. Ann. Appl. Probab. 18 (2008) 1026-1058. | MR 2418237 | Zbl 1141.60068

[23] V. Vysotsky. On the probability that integrated random walks stay positive. Stochastic Process. Appl. 120 (2010) 1178-1193. | MR 2639743 | Zbl 1202.60070

[24] V. M. Zolotarev. One-Dimensional Stable Distributions. Amer. Math. Soc., Providence, RI, 1986. | MR 854867 | Zbl 0589.60015