Kolmogorov’s law of the iterated logarithm for noncommutative martingales
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1124-1130.

Nous prouvons la loi de Kolmogorov du logarithme itéré pour des martingales non-commutatives. Le cas commutatif a été établi par Stout. L’ingrédient clé est une inégalité exponentielle prouvée récemment par Junge et l’auteur.

We prove Kolmogorov’s law of the iterated logarithm for noncommutative martingales. The commutative case was due to Stout. The key ingredient is an exponential inequality proved recently by Junge and the author.

DOI : https://doi.org/10.1214/14-AIHP603
Mots clés : law of the iterated logarithm, noncommutative martingales, quantum martingales, exponential inequality
@article{AIHPB_2015__51_3_1124_0,
     author = {Zeng, Qiang},
     title = {Kolmogorov's law of the iterated logarithm for noncommutative martingales},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1124--1130},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {3},
     year = {2015},
     doi = {10.1214/14-AIHP603},
     mrnumber = {3365975},
     language = {en},
     url = {archive.numdam.org/item/AIHPB_2015__51_3_1124_0/}
}
Zeng, Qiang. Kolmogorov’s law of the iterated logarithm for noncommutative martingales. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1124-1130. doi : 10.1214/14-AIHP603. http://archive.numdam.org/item/AIHPB_2015__51_3_1124_0/

[1] H. Bauer. Probability Theory. de Gruyter Studies in Mathematics 23. Walter de Gruyter, Berlin, 1996. Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author. | MR 1385460 | Zbl 0868.60001

[2] A. De Acosta. A new proof of the Hartman–Wintner law of the iterated logarithm. Ann. Probab. 11 (2) (1983) 270–276. | MR 690128 | Zbl 0512.60014

[3] T. Fack and H. Kosaki. Generalized s-numbers of τ-measurable operators. Pacific J. Math. 123 (2) (1986) 269–300. | MR 840845 | Zbl 0617.46063

[4] P. Hartman and A. Wintner. On the law of the iterated logarithm. Amer. J. Math. 63 (1941) 169–176. | JFM 67.0460.03 | MR 3497

[5] R. Imbuzeiro Oliveira. Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges. ArXiv e-prints, 2009.

[6] M. Junge. Doob’s inequality for non-commutative martingales. J. Reine Angew. Math. 549 (2002) 149–190. | MR 1916654 | Zbl 1004.46043

[7] M. Junge and Q. Xu. Noncommutative Burkholder/Rosenthal inequalities. Ann. Probab. 31 (2) (2003) 948–995. | MR 1964955 | Zbl 1041.46050

[8] M. Junge and Q. Xu. Noncommutative maximal ergodic theorems. J. Amer. Math. Soc. 20 (2) (2007) 385–439. | MR 2276775 | Zbl 1116.46053

[9] M. Junge and Q. Zeng. Noncommutative martingale deviation and Poincaré type inequalities with applications. Preprint, 2012. | MR 3334274

[10] M. Konwerska. The Law of the Iterated Logarithm in Noncommutative Probability. ProQuest LLC, Ann Arbor, MI, 2008. Ph.D. Thesis, Univ. Illinois at Urbana–Champaign. | MR 2712889

[11] M. Konwerska. The law of the iterated logarithm in noncommutative probability. Preprint, 2012. | MR 2712889

[12] M. Ledoux and M. Talagrand. Probability in Banach Spaces: Isoperimetry and Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 23. Springer, Berlin, 1991. | MR 1102015 | Zbl 0748.60004

[13] G. Pisier. Non-commutative vector valued L p -spaces and completely p-summing maps. Astérisque 247 (1998) vi+131. | MR 1648908 | Zbl 0937.46056

[14] G. Pisier and Q. Xu. Non-commutative L p -spaces. In Handbook of the Geometry of Banach Spaces, Vol. 2 1459–1517. North-Holland, Amsterdam, 2003. | MR 1999201 | Zbl 1046.46048

[15] W. F. Stout. A martingale analogue of Kolmogorov’s law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 15 (1970) 279–290. | MR 293701 | Zbl 0209.49004

[16] W. F. Stout. The Hartman–Wintner law of the iterated logarithm for martingales. Ann. Math. Statist. 41 (1970) 2158–2160. | Zbl 0235.60046

[17] M. Terp. L p -spaces associated with von Neumann algebras. Notes, Copenhagen Univ., 1981.

[18] D. V. Voiculescu, K. J. Dykema and A. Nica. Free Random Variables. CRM Monograph Series 1. Amer. Math. Soc., Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. | MR 1217253 | Zbl 0795.46049

[19] W. B. Wu. Strong invariance principles for dependent random variables. Ann. Probab. 35 (6) (2007) 2294–2320. | MR 2353389 | Zbl 1166.60307

[20] O. Zhao and M. Woodroofe. Law of the iterated logarithm for stationary processes. Ann. Probab. 1 (2008) 127–142. | MR 2370600 | Zbl 1130.60039