Soit la somme partielle itérée, c’est à dire , où . Pour des variables aléatoires i.i.d. intégrables et de moyenne nulle, nous montrons que les probabilités de persistance satisfont
Let denote the iterated partial sums. That is, , where . Assuming are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities
Mots-clés : first passage time, iterated partial sums, persistence, lower tail probability, one-sided probability, random walk
@article{AIHPB_2013__49_3_873_0, author = {Dembo, Amir and Ding, Jian and Gao, Fuchang}, title = {Persistence of iterated partial sums}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {873--884}, publisher = {Gauthier-Villars}, volume = {49}, number = {3}, year = {2013}, doi = {10.1214/11-AIHP452}, mrnumber = {3112437}, zbl = {1274.60144}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP452/} }
TY - JOUR AU - Dembo, Amir AU - Ding, Jian AU - Gao, Fuchang TI - Persistence of iterated partial sums JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2013 SP - 873 EP - 884 VL - 49 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP452/ DO - 10.1214/11-AIHP452 LA - en ID - AIHPB_2013__49_3_873_0 ER -
%0 Journal Article %A Dembo, Amir %A Ding, Jian %A Gao, Fuchang %T Persistence of iterated partial sums %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 873-884 %V 49 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP452/ %R 10.1214/11-AIHP452 %G en %F AIHPB_2013__49_3_873_0
Dembo, Amir; Ding, Jian; Gao, Fuchang. Persistence of iterated partial sums. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 873-884. doi : 10.1214/11-AIHP452. http://www.numdam.org/articles/10.1214/11-AIHP452/
[1] Survival probabilities of weighted random walks. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011) 235-258. | MR | Zbl
and .[2] Universality of the asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 236-251. | Numdam | MR
and .[3] Pinning and wetting transition for -dimensional fields with Laplacian interaction. Ann. Probab. 36 (2008) 2388-2433. | MR | Zbl
and .[4] Random polynomials having few or no real zeros. J. Amer. Math. Soc. 15 (2002) 857-892. | MR | Zbl
, , and .[5] The lower tail problem for the area of a symmetric stable process. Unpublished manuscript, 2007.
, and .[6] An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York, 1971. | MR | Zbl
.[7] Recent developments on lower tail probabilities for Gaussian processes. Cosmos 1 (2005) 95-106. | MR
and .[8] A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227-235. | MR | Zbl
[9] Comparison of sums of independent identically distributed random vectors. Probab. Math. Statist. 14 (1993) 281-285. | MR | Zbl
.[10] The lower tail problem for homogeneous functionals of stable processes with no negative jumps. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007) 165-179. | MR | Zbl
.[11] Distribution of some functionals of the integral of a random walk. Theoret. Math. Phys. 90 (1992) 219-241. | MR | Zbl
.[12] Clustering in a stochastic model of one-dimensional gas. Ann. Appl. Probab. 18 (2008) 1026-1058. | MR | Zbl
.[13] On the probability that integrated random walks stay positive. Stochastic Process. Appl. 120 (2010) 1178-1193. | MR | Zbl
.Cité par Sources :