An exact asymptotic for the square variation of partial sum processes

Lewko, Allison ; Lewko, Mark

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 1597-1619 / Harvested from Numdam

Résumé

Nous établissons une formule asymptotique exacte pour la variation quadratique de certains processus de sommes partielles. Soit $\lbrace X_{i}\rbrace $ une suite de variables indépendantes et identiquement distribuées de moyenne nulle et de variance finie $\sigma ^{2}$ satisfaisant une condition de moments $\mathbb {E}[|X_{i}|^{2+\delta }]<\infty $ pour un $\delta >0$. Soit $\mathcal {P}_{N}$ l’ensemble de toutes les partitions possibles de l’intervalle $[N]$ en sous-intervalles, alors nous montrons que presque sûrement $\max _{\pi \in \mathcal {P}_{N}}\sum _{I\in \pi }|\sum _{i\in I}X_{i}|^{2}\sim 2\sigma ^{2}N\ln \ln (N)$. Ceci peut être interprété comme une amélioration de la loi du logarithme itéré et précise les résultats de J. Qian sur les sommes partielles et les processus empiriques. Quand $\delta =0$, nous obtenons une version plus faible, en probabilité, de ce résultat.

Abstract

We establish an exact asymptotic formula for the square variation of certain partial sum processes. Let $\lbrace X_{i}\rbrace $ be a sequence of independent, identically distributed mean zero random variables with finite variance $\sigma ^{2}$ and satisfying a moment condition $\mathbb {E}[|X_{i}|^{2+\delta }]<\infty $ for some $\delta >0$. If we let $\mathcal {P}_{N}$ denote the set of all possible partitions of the interval $[N]$ into subintervals, then we have that $\max _{\pi \in \mathcal {P}_{N}}\sum _{I\in \pi }|\sum _{i\in I}X_{i}|^{2}\sim 2\sigma ^{2}N\ln \ln (N)$ holds almost surely. This can be viewed as a variational strengthening of the law of the iterated logarithm and refines results of J. Qian on partial sum and empirical processes. When $\delta =0$, we obtain a weaker ‘in probability’ version of the result.

Détail

Publié le : 2015-01-01

MR 3414459

DOI : https://doi.org/10.1214/14-AIHP617

Bibliographie

[1] J. Bretagnolle. $p$-variation de fonctions aléatoires. II. Processus à accroissements indépendants. In Séminaire de Probabilités VI (Univ. Strasbourg, Année Universitaire 1970–1971). Lecture Notes in Math. 258 64–71. Springer, Berlin, 1972. | Numdam | MR 373020 | Zbl 0273.60051

[2] Y. S. Chow and H. Teicher. Probability Theory: Independence, Interchangeability, Martingales, 3rd edition. Springer, Berlin, 1997. | MR 1476912 | Zbl 0399.60001

[3] J. L. Doob. Stochastic Processes. Wiley, New York, 1953. | MR 58896 | Zbl 0696.60003

[4] R. M. Dudley. Empirical processes and $p$-variation. In Festschrift for Lucien Le Cam 219–233. Springer, Berlin, 1997. | MR 1462947 | Zbl 0914.62037

[5] N. Etemadi. On some classical results in probability theory. Sankhyā Ser. A 47 (1985) 215–221. | MR 844022 | Zbl 0581.60019

[6] J. Galambos. Advanced Probability Theory, 2nd edition. Dekker, New York, 1995. | MR 1350792 | Zbl 0841.60001

[7] 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

[8] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (301) (1963) 13–30. | MR 144363 | Zbl 0127.10602

[9] R. Jones and G. Wang. Variation inequalities for the Fejér and Poisson kernels. Trans. Amer. Math. Soc. 356 (11) (2004) 4493–4518. | MR 2067131 | Zbl 1065.42006

[10] A. Lewko and M. Lewko. Estimates for the square variation of partial sums of Fourier series and their rearrangements. J. Funct. Anal. 262 (6) (2012) 2561–2607. | MR 2885959 | Zbl 1239.42027

[11] A. Lewko and M. Lewko. Orthonormal systems in linear spans. Anal. PDE 7 (2014) 97–115. | MR 3219501 | Zbl 1298.42032

[12] A. Lewko and M. Lewko. A variational Barban–Davenport–Halberstam theorem. J. Number Theory 132 (9) (2012) 2020–2045. | MR 2925860 | Zbl 06052917

[13] A. Lewko and M. Lewko. The square variation of rearranged Fourier series. Amer. J. Math. To appear, 2015. Available at arXiv:1212.1988. | MR 3405868

[14] J. Qian. The $p$-variation of partial sum processes and the empirical process. Ann. Probab. 26 (3) (1998) 1370–1383. | MR 1640349 | Zbl 0927.62049

[15] E. Rio. Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques and Applications 31. Springer, Berlin, 1999. | Zbl 0944.60008

[16] H. P. Rosenthal. On the subspaces of $L^{p}$ ($p>2$) spanned by sequences of independent random variables. Israel J. Math. 8 (1970) 273–303. | MR 271721 | Zbl 0213.19303

[17] S. J. Taylor. Exact asymptotic estimates of Brownian path variation. Duke Math. J. 39 (1972) 219–241. | MR 295434 | Zbl 0241.60069