Abstract
This paper presents a sharp bound on the L p norm (1 ≤ p ≤ 2) of a randomly stopped multilinear form of i.i.d. mean zero random variables. As a corollary we obtain optimal conditions for Wald’s equation for this multilinear form to hold. The bound obtained generalizes earlier work of (1991) and (1988). The techniques used include decoupling inequalities and the argument of subsequences.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Blackwell, D. (1946). On an equation of Wald. Ann. Math. Stat. 17, 84–87.
Burkholder, D. L. (1966). Martingale transforms. Ann. Math. Stat. 37, 1494–1504.
Burkholder, D. and Gundy, R. (1970). Extrapolation and interpolation of quasilinear operators on martingales. Acta Math. 124, 249–304.
Chow, Y.S., Robbins, H. and Siegmund, D. (1971). Great Expectations. The Theory of Optimal Stopping. Houghton Mifflin, New York.
Chow, Y.S., de la Pena, V.H., Teicher, H. (1991). Wald’s equation for a class of de-normalized U-statistics. To appear in Ann. Probab.
Davis, B. (1970). On the integrability of the martingale square function. Israel J. of Math. 8, 187–190.
de la Peña, V.H. (1987). L p Bounds for quadratic forms (preprint).
de la Peña, V.H. (1988). L-Bounds of best possible type for martingales, degenerate U-statistics, and certain linear forms. PhD. Thesis. Statistics Department, University of California, Berkeley.
de la Peña, V.H. (1990). Decoupling and Khintchine’s inequalities for U-statistics. To appear in Ann. Probab.
de la Pena, V.H. and Klass, M.J. (1990) Order of magnitude bounds for expectations involving quadratic forms. To appear in Ann. Probab.
Klass, M.J. (1980). Precision bounds for the relative error in the approximation of E∣S n∣ and extensions. Ann. Probab. 8(2), 350–367.
Klass, M.J. (1981). A method of approximating expectations of functions of sums of independent random variables. Ann. Probab. 9(3), 413–428.
Klass, M.J. (1988). A best possible improvement of Wald’s equation. Ann. Probab. 16, 840–853.
Wald, A. (1945). Sequential analysis of statistical hypotheses. Ann. Math. Stat. 16, 117–186.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media New York
About this chapter
Cite this chapter
de la Peña, V.H. (1992). Sharp Bounds on the LP Norm of a Randomly Stopped Multilinear form with an Application to Wald’s Equation. In: Dudley, R.M., Hahn, M.G., Kuelbs, J. (eds) Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Progress in Probability, vol 30. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0367-4_4
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0367-4_4
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6728-7
Online ISBN: 978-1-4612-0367-4
eBook Packages: Springer Book Archive