An Exact Asymptotic for the Square Variation of Partial Sum Processes

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 $\max_{\pi \in \mathcal{P}_{N}} \sum_{I \in \pi} | \sum_{i\in I} X_{i}|^2 \sim 2 \sigma^2N \ln \ln(N)$ 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.