The Annals of Probability

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

Jinghua Qian

Full-text: Open access

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.

Article information

Source
Ann. Probab., Volume 26, Number 3 (1998), 1370-1383.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855756

Digital Object Identifier
doi:10.1214/aop/1022855756

Mathematical Reviews number (MathSciNet)
MR1640349

Zentralblatt MATH identifier
0927.62049

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

Keywords
$p$-variation norm $\psi$-variation the empirical process partial sum processes

Citation

Qian, Jinghua. The $p$-variation of partial sum processes and the empirical process. Ann. Probab. 26 (1998), no. 3, 1370--1383. doi:10.1214/aop/1022855756. https://projecteuclid.org/euclid.aop/1022855756


Export citation

References

  • [1] Breiman,L. (1968). Probability. Addison-Wesley, Reading, MA.
  • [2] Bretagnolle,J. (1972). p-variation de fonctions al´eatoires. II. Processus a accroissements ind´ependants. Lecture Notes in Math. 258 64-71. Springer, Berlin.
  • [3] Chung,K. L. (1949). An estimate concerning the Kolmogoroff limit distribution. Trans. Amer. Math. Soc. 67 36-50.
  • [4] Dudley,R. M. (1992). Fr´echet differentiability, p-variation and uniform Donsker classes. Ann. Probab. 20 1968-1982.
  • [5] Dudley,R. M. (1993). Real Analysis and Probability, 2nd ed. Chapman and Hall, New York.
  • [6] Dudley,R. M. (1994). The order of the remainder in derivatives of composition and inverse operators for p-variation norms. Ann. Statist. 22 1-20.
  • [7] Dudley,R. M. (1997). Empirical processes and p-variation. In Festschrift for Lucien Le Cam (D. Pollard, E. Torgersen and G. L. Yang, eds.) 219-233. Springer, New York.
  • [8] Dudley,R. M. and Philipp,W. (1983). Invariance principles for sums of Banach space valued random elements and empirical processes.Wahrsch. Verw. Gebiete 62 509- 552.
  • [9] Huang,Y.-C. (1994). Speed of convergence of classical empirical processes in p-variationnorm. Ph.D. dissertation, MIT.
  • [10] Kuelbs,J. (1977). Kolmogorov's law of the iterated logarithm for Banach space valued random variables. Illinois J. Math. 21 784-800.
  • [11] L´epingle,D. (1976). La variation d'ordre p des semi-martingales.Wahrsch. Verw. Gebiete 36 295-316.
  • [12] Love,E. R. and Young,L. C. (1937). Sur une classe de fonctionelles lin´eaires. Fund. Math 28 243-257.
  • [13] Petrov,V. V. (1975). Sums of Independent Random Variables. Springer, New York.
  • [14] Pisier,G. and Xu,Q. (1988). The strong p-variation of martingales and orthogonal series. Probab. Theory Related Fields 77 497-514.
  • [15] Qian,J. (1997). The p-variation of partial sum processes and empirical processes. Ph.D. dissertation, Tufts Univ.
  • [16] Shorack,G. R. and Wellner,J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • [17] Smirnov,N. V. (1944). Approximate laws of distribution of random variables from empirical data. Uspekhi Mat. Nauk 10 179-206 (in Russian).
  • [18] Taylor,S. J. (1972). Exact asymptotic estimates of Brownian path variation. Duke Math. J. 39 219-241.
  • [19] Wiener,N. (1924). The quadratic variation of a function and its Fourier coefficients. J. Math. Phys. 3 72-94.
  • [20] Young,L. C. (1936). An inequality of the H¨older type, connected with Stieltjes integration. Acta Math. 67 251-282.