An Exact Asymptotic for the Square Variation of Partial Sum Processes

TL;DR

This paper derives an exact asymptotic formula for the maximum square variation of partial sum processes of i.i.d. mean-zero variables, refining the law of the iterated logarithm with a variational approach.

Contribution

It establishes a precise asymptotic behavior for the square variation of partial sums, extending classical results to a variational setting with almost sure convergence.

Findings

Max square variation asymptotically behaves as 2σ²N ln ln(N).

Results refine the law of the iterated logarithm for partial sums.

Provides a weaker in-probability version when δ=0.

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