An Exact Asymptotic for the Square Variation of Partial Sum Processes

Abstract

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 $\sigma$ 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.

23 pages