## The Annals of Probability

### On extremal distributions and sharp $\bolds{L_p}$-bounds\\ for sums of multilinear forms

#### Abstract

In this paper we present a study of the problem of approximating the expectations of functions of statistics in independent and dependent random variables in terms of the expectations of functions of the component random variables. We present results providing sharp analogues of the Burkholder--Rosenthal inequalities and related estimates for the expectations of functions of sums of dependent nonnegative r.v.'s and conditionally symmetric martingale differences with bounded conditional moments as well as for sums of multilinear forms. Among others, we obtain the following sharp inequalities: $E(\sum_{k=1}^n X_k)^t\le 2 \max (\sum_{k=1}^n EX_k^t, (\sum_{k=1}^n a_k)^t)$ for all nonnegative r.v.'s $X_1, \ldots, X_n$ with $E(X_k\mid X_1, \ldots, X_{k-1})\le a_k$, $EX_k^t<\infty$, $k=1, \ldots, n$, $1#x003C;t#x003C;2$; $E(\sum_{k=1}^n X_k)^t\le E\theta^t(1) \max (\sum_{k=1}^n b_k, (\sum_{k=1}^n a_k^s)^{t/s})$ for all nonnegative r.v.'s $X_1, \ldots, X_n$ with $E(X_k^s\mid X_1, \ldots, X_{k-1})\le a_k^s$, $E(X_k^t\mid X_1, \ldots, X_{k-1})\le b_k$, $k=1, \ldots, n$, $1#x003C;t#x003C;2$, $0#x003C;s\le t-1$ or $t\ge 2$, $0#x003C;s\le 1$, where $\theta(1)$ is a Poisson random variable with parameter 1. As applications, new decoupling inequalities for sums of multilinear forms are presented and sharp Khintchine--Marcinkiewicz--Zygmund inequalities for generalized moving averages are obtained. The results can also be used in the study of a wide class of nonlinear statistics connected to problems of long-range dependence and in an econometric setup, in particular, in stabilization policy problems and in the study of properties of moving average and autocorrelation processes. The results are based on the iteration of a series of key lemmas that capture the essential extremal properties of the moments of the statistics involved.

#### Article information

Source
Ann. Probab. Volume 31, Number 2 (2003), 630-675.

Dates
First available: 24 March 2003

http://projecteuclid.org/euclid.aop/1048516531

Digital Object Identifier
doi:10.1214/aop/1048516531

Mathematical Reviews number (MathSciNet)
MR1964944

Zentralblatt MATH identifier
1033.60019

#### Citation

de la Peña, Victor H.; Ibragimov, Rustam; Sharakhmetov, Shaturgun. On extremal distributions and sharp $\bolds{L_p}$-bounds\\ for sums of multilinear forms. The Annals of Probability 31 (2003), no. 2, 630--675. doi:10.1214/aop/1048516531. http://projecteuclid.org/euclid.aop/1048516531.

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• NEW YORK, NEW YORK 10027 E-MAIL: vp@stat.columbia.edu R. IBRAGIMOV DEPARTMENT OF ECONOMICS YALE UNIVERSITY 28 HILLHOUSE AVENUE
• NEW HAVEN, CONNECTICUT 06511 E-MAIL: rustam.ibragimov@yale.edu SH. SHARAKHMETOV DEPARTMENT OF PROBABILITY THEORY TASHKENT STATE ECONOMICS UNIVERSITY UL. UZBEKISTANSKAy A, 49 TASHKENT 700063 UZBEKISTAN E-MAIL: tim001@tseu.silk.org