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On Extremal Distributions and Sharp L[sub]p-Bounds For Sums of Multilinear Forms

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  • de la Peña, Victor H.
  • Sharakhmetov, Shaturgun
  • Ibragimov, Rustam

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

Suggested Citation

  • de la Peña, Victor H. & Sharakhmetov, Shaturgun & Ibragimov, Rustam, 2003. "On Extremal Distributions and Sharp L[sub]p-Bounds For Sums of Multilinear Forms," Scholarly Articles 2624455, Harvard University Department of Economics.
    • Handle: RePEc:hrv:faseco:2624455
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    1. R. Ibragimov & Sh. Sharakhmetov, 1999. "Analogues of Khintchine, Marcinkiewicz–Zygmund and Rosenthal Inequalities for Symmetric Statistics," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(4), pages 621-633, December.
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    Cited by:

    1. Donald Brown & Rustam Ibragimov & Johan Walden, 2015. "Bounds for path-dependent options," Annals of Finance, Springer, vol. 11(3), pages 433-451, November.

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