## 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

### Citations:

Zbl 0778.60026; Zbl 0816.62039; Zbl 0914.62037; Zbl 0273.60052
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### References:

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