dc.creator de la Peña, Victor H. dc.creator Sharakhmetov, Shaturgun dc.creator Ibragimov, Rustam dc.date 2009-02-20T07:46:24Z dc.date 2003 dc.date.accessioned 2012-06-07T21:42:57Z dc.date.available 2012-06-07T21:42:57Z dc.date.issued 2012-06-07 dc.identifier de 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.identifier 0091-1798 dc.identifier http://nrs.harvard.edu/urn-3:HUL.InstRepos:2624455 dc.identifier.uri https://repositorio.leon.uia.mx/xmlui/123456789/33152 dc.description 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. dc.description Economics dc.language en_US dc.publisher The Institute of Mathematical Statistics dc.relation http://dx.doi.org/10.1214/aop/1048516531 dc.relation Annals of Probability dc.subject sums of multilinear forms dc.subject Burkholder-Rosenthal-type and Khintchine-type inequalities dc.subject statistics dc.subject autocorrelation processes dc.subject stochastic Taylor expansion dc.subject extremal distributions dc.subject decoupling inequalities dc.subject moving average processes dc.subject long-range dependence dc.subject nonlinear statistics dc.title On Extremal Distributions and Sharp L[sub]p-Bounds For Sums of Multilinear Forms.
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