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We establish an exact asymptotic formula for the square variation of certainpartial 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 ofall possible partitions of the interval $[N]$ into subintervals, then we havethat $\max_{\pi \in \mathcal{P}_{N}} \sum_{I \in \pi} | \sum_{i\in I} X_{i}|^2\sim 2 \sigma^2N \ln \ln(N)$ holds almost surely. This can be viewed as avariational strengthening of the law of the iterated logarithm and refinesresults of J. Qian on partial sum and empirical processes. When $\delta = 0$,we obtain a weaker `in probability' version of the result.

An Exact Asymptotic for the Square Variation of Partial Sum Processes