Positivity of integrated random walks

Vysotsky, Vladislav

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 195-213.

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 $\alpha$-stable avec $1<\alpha \le 2$. En supposant que $S_1$ est soit exponentielle à droite (i.e. $\mathbb{P}(S_1>x|S_1>0)={\mathrm{e}}^{-ax}$), soit continue à droite (i.e. $\mathbb{P}(S_1=1|S_1>0)=1$), nous prouvons que

$$\mathbb{P}\{A_1>0,\cdots ,A_N>0\}\sim C_{\alpha }{N}^{1/\left(2\alpha \right)-1/2}$$

quand $N\to \infty$, où $C_{\alpha }>0$ 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 $\alpha$-stable law with $1<\alpha \le 2$. Assuming that $S_1$ is either right-exponential (i.e. $\mathbb{P}(S_1>x|S_1>0)={\mathrm{e}}^{-ax}$ for some $a>0$ and all $x>0$) or right-continuous (skip free), we prove that

$$\mathbb{P}\{A_1>0,\cdots ,A_N>0\}\sim C_{\alpha }{N}^{1/\left(2\alpha \right)-1/2}$$

as $N\to \infty$, where $C_{\alpha }>0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Classification : 60G50, 60F99
Mots clés : integrated random walk, persistence, one-sided exit probability, unilateral small deviations, area of random walk, Sparre-Andersen theorem, stable excursion, area of excursion

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

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

[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

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

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

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

[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 | Zbl

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

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

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