An exact asymptotic for the square variation of partial sum processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, p. 1597-1619
Nous établissons une formule asymptotique exacte pour la variation quadratique de certains processus de sommes partielles. Soit {X i } une suite de variables indépendantes et identiquement distribuées de moyenne nulle et de variance finie σ 2 satisfaisant une condition de moments 𝔼[|X i | 2+δ ]< pour un δ>0. Soit 𝒫 N l’ensemble de toutes les partitions possibles de l’intervalle [N] en sous-intervalles, alors nous montrons que presque sûrement max π𝒫 N Iπ | iI X i | 2 2σ 2 Nlnln(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 δ=0, nous obtenons une version plus faible, en probabilité, de ce résultat.
We establish an exact asymptotic formula for the square variation of certain partial sum processes. Let {X i } be a sequence of independent, identically distributed mean zero random variables with finite variance σ 2 and satisfying a moment condition 𝔼[|X i | 2+δ ]< for some δ>0. If we let 𝒫 N denote the set of all possible partitions of the interval [N] into subintervals, then we have that max π𝒫 N Iπ | iI X i | 2 2σ 2 Nlnln(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 δ=0, we obtain a weaker ‘in probability’ version of the result.
@article{AIHPB_2015__51_4_1597_0,
     author = {Lewko, Allison and Lewko, Mark},
     title = {An exact asymptotic for the square variation of partial sum processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {4},
     year = {2015},
     pages = {1597-1619},
     doi = {10.1214/14-AIHP617},
     mrnumber = {3414459},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_4_1597_0}
}
Lewko, Allison; Lewko, Mark. An exact asymptotic for the square variation of partial sum processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1597-1619. doi : 10.1214/14-AIHP617. https://www.numdam.org/item/AIHPB_2015__51_4_1597_0/

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