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The \(p\)-variation of partial sum processes and the empirical process. (English) Zbl 0927.62049

Summary: 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. R. M. Dudley [Ann. Probab. 20, No. 4, 1968-1982 (1992; Zbl 0778.60026); Ann. Stat. 22, No. 1, 1-20 (1994; Zbl 0816.62039); D. Pollard et al. (eds.), Festschrift for Lucien Le Cam, 219-233 (1997; Zbl 0914.62037)] 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. 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 \(S_j:=X_1+X_2+\cdots+X_j\). 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 \(j-1<t\leq j\), \(j=1,2,\dots,n\). J. Bretagnolle [Lecture Notes Math. 258, 64-71 (1972; Zbl 0273.60052)] 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.

MSC:

62G30 Order statistics; empirical distribution functions
60G50 Sums of independent random variables; random walks
26A45 Functions of bounded variation, generalizations
26A48 Monotonic functions, generalizations
62G20 Asymptotic properties of nonparametric inference
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