## On extremal distributions and sharp $$L_p$$-bounds for sums of multilinear forms.(English)Zbl 1033.60019

Authors’ abstract: We present a study of the problem of approximating the expectations of functions of statistics in independent and dependent 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: $E\left(\sum_{k=1}^nX_k\right)^t \leq 2\max\left(\sum_{k=1}^n EX_k^t,\left(\sum_{k=1}^na_k\right)^t\right)$ for all nonnegative r.v.’s $$X_1,\dots,X_n$$ with $$E(X_k\mid X_1,\dots,X_{k-1})\leq a_k$$, $$EX_k^t<\infty$$, $$k=1,\dots,n$$, $$1<t<2$$; $$E(\sum_{k=1}^nX_k)^t \leq E(\theta^t(1))\max(\sum_{k=1}^nb_k, (\sum_{k=1}^na_k^s)^{t/s})$$ for all nonnegative r.v.’s $$X_1, \dots,X_n$$ with $$E(X_k^s\mid X_1,\dots,X_{k-1})\leq a_k^s$$, $$E(X_k^t\mid X_1,\dots,X_{k-1})\leq b_k$$, $$k=1,\dots,n$$, $$1<s\leq t-1$$ or $$t\geq 2$$, $$0< s\leq 1$$, where $$\theta(1)$$ is a random Poisson random variable with parameter 1. As applications, new decoupling inequalities for sums of multilinear forms are presented and sharp Khinchin-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 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.

### MSC:

 60E15 Inequalities; stochastic orderings 60F25 $$L^p$$-limit theorems 60G50 Sums of independent random variables; random walks
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### References:

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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
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