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) no. 4, pp. 1597-1619.

Nous établissons une formule asymptotique exacte pour la variation quadratique de certains processus de sommes partielles. Soit $\left\{X_i\right\}$ une suite de variables indépendantes et identiquement distribuées de moyenne nulle et de variance finie ${\sigma }^2$ satisfaisant une condition de moments $𝔼\left[\right|X_i{|}^{2+\delta }]<\infty$ pour un $\delta >0$. Soit ${𝒫}_N$ l’ensemble de toutes les partitions possibles de l’intervalle $\left[N\right]$ en sous-intervalles, alors nous montrons que presque sûrement ${max}_{\pi \in {𝒫}_N}{\sum }_{I\in \pi }{\left|{\sum }_{i\in I}{X}_i\right|}^2\sim 2{\sigma }^{2}Nlnln\left(N\right)$. 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.

We establish an exact asymptotic formula for the square variation of certain partial sum processes. Let $\left\{X_i\right\}$ be a sequence of independent, identically distributed mean zero random variables with finite variance ${\sigma }^2$ and satisfying a moment condition $𝔼\left[\right|X_i{|}^{2+\delta }]<\infty$ for some $\delta >0$. If we let ${𝒫}_N$ denote the set of all possible partitions of the interval $\left[N\right]$ into subintervals, then we have that ${max}_{\pi \in {𝒫}_N}{\sum }_{I\in \pi }{\left|{\sum }_{i\in I}{X}_i\right|}^2\sim 2{\sigma }^{2}Nlnln\left(N\right)$ 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.

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