The $p$-variation of partial sum processes and the empirical process

Abstract

The $p$-variation of a function $f$ is the supremum of the sums of the $p$th powers of absolute increments of f over nonoverlapping intervals. Let $F$ be a continuous probability distribution function. Dudley has shown that the $p$-variation of the empirical process is bounded in probability as $n \to \infty$ if and only if $p > 2$, and for $1 \leq p \leq 2$, the $p$-variation of the empirical process is at least $n^{1-p/2}$ and is at most of the order $n^{1-p/2}(\log \log n)^{p/2}$ in probability. In this paper, we prove that the exact order of the 2-variation of the empirical process is $\log \log n$ in probability, and for $1 \leq p < 2$, the $p$-variation of the empirical process is of exact order $n^{1-p/2}$ in expectation and almost surely.

Let $S_j := X_1 + X_2 + \dots + X_j$. Then the p-variation of the partial sum process for ${X_1, X_2, \dots, X_n}$ is defined as that of f on $(0, n]$, where $f(t) = S_j$ for $j - 1 < t \leq j, j = 1, 2, \dots, n$. Bretagnolle has shown that the expectation of the $p$-variation for independent centered random variables $X_i$ with bounded $p$th moments is of order $n$ for $1 \leq p < 2$. We prove that for $p = 2$, the 2-variation of the partial sum process of i.i.d. centered nonconstant random variables with finite $2 + \delta$ moment for some $\delta > 0$ is of exact order $n \log \log n$ in probability.