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dc.creatorde la Peña, Victor H.
dc.creatorSharakhmetov, Shaturgun
dc.creatorIbragimov, Rustam
dc.identifierde la Peña, Victor H., Rustam Ibragimov and Shaturgun Sharakhmetov. 2003. On extremal distributions and sharp L[sub]p-bounds for sums of multilinear forms. Annals of Probability 31(2): 630-675.
dc.descriptionIn 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.
dc.publisherThe Institute of Mathematical Statistics
dc.relationAnnals of Probability
dc.subjectsums of multilinear forms
dc.subjectBurkholder-Rosenthal-type and Khintchine-type inequalities
dc.subjectautocorrelation processes
dc.subjectstochastic Taylor expansion
dc.subjectextremal distributions
dc.subjectdecoupling inequalities
dc.subjectmoving average processes
dc.subjectlong-range dependence
dc.subjectnonlinear statistics
dc.titleOn Extremal Distributions and Sharp L[sub]p-Bounds For Sums of Multilinear Forms.

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