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Order of magnitude bounds for expectations of $$\Delta_2$$-functions of nonnegative random bilinear forms and generalized $$U$$-statistics. (English) Zbl 0895.60018
Summary: Let $$X_1,Y_1,X_2,Y_2, \dots, X_n,Y_n$$ be independent nonnegative rv’s and let $$\{b_{ij}\}_{1\leq i,j\leq n}$$ be an array of nonnegative constants. We present a method of obtaining the order of magnitude of
$$E\Phi (\sum_{1\leq i,j \leq n} b_{ij} X_iY_j)$$, for any such $$\{X_i\}$$, $$\{Y_j\}$$ and $$\{b_{ij}\}$$ and any nondecreasing function $$\Phi$$ on $$[0,\infty)$$ with $$\Phi (0)=0$$ and satisfying a $$\Delta_2$$ growth condition. Furthermore, this technique is extended to provide the order of magnitude of $$E\Phi (\sum_{1\leq i,j \leq n} f_{ij} (X_i,Y_j))$$, where $$\{f_{ij} (x,y)\}_{1\leq i,j \leq n}$$ is any array of nonnegative functions.
For arbitrary functions $$\{ g_{ij} (x,y)\}_{1\leq i\neq j\leq n}$$, the aforementioned approximation enables us to identify the order of magnitude of $$E\Phi (|\sum_{1\leq i\neq j\leq n} g_{ij} (X_i,X_j)|)$$ whenever decoupling results and Khinchin-type inequalities apply, such as $$\Phi$$ is convex, $${\mathcal L}(g_{ij} (X_i, X_j))={\mathcal L}(g_{ji} (X_j,X_i))$$ and $$Eg_{ij} (X_i,x) \equiv 0$$ for all $$x$$ in the range of $$X_j$$.

##### MSC:
 60E15 Inequalities; stochastic orderings 60F25 $$L^p$$-limit theorems 60G50 Sums of independent random variables; random walks
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##### References:
 [1] BOURGAIN, J. and TZAFRIRI, L. 1987. Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis. Israel J. Math. 57 137 224. Z. · Zbl 0631.46017 [2] CAMBANIS, S., ROSINSKI, J. and WOYCZYNSKI, W. A. 1985. Convergence of quadratic forms in \' ṕ-stable random variables and -radonifying operators. Ann. Probab. 13 885 897. p Z. · Zbl 0575.60018 [3] DE LA PENA, V. H. 1990. Bounds on the expectation of functions of martingales and sums of positive R.V.’s in terms of norms of sums of independent random variables. Proc. Amer. Math. Soc. 108 233 239. Z. · Zbl 0682.60032 [4] DE LA PENA, V. H. and KLASS, M. J. 1994. Order-of-magnitude bounds for expectations involving quadratic forms. Ann. Probab. 22 1044 1077.Z. · Zbl 0810.60012 [5] DE LA PENA, V. H. and MONTGOMERY-SMITH, S. J. 1995. Decoupling inequalities for tail probabilities of multilinear forms of multivariate U-statistics. Ann. Probab. 23 806 816. Z. · Zbl 0827.60014 [6] GINE, E. and ZINN, J. 1992. On Hoffmann-Jørgensen’s inequality for U-processes. In Probabilíty in Banach Spaces 8. Progress in Probability 30 80 91. Birkhauser, Boston. \" Z. · Zbl 0787.60020 [7] HALL, P. and HEYDE, C. C. 1980. Martingale Limit Theory and Its Applications, 2nd ed. Academic Press, New York. Z. HOFFMANN-J øRGENSEN, J. 1974. Sums of independent Banach space valued random variables. Studia Math. 52 159 186. [8] KAHANE, J.-P. 1985. Some Random Series of Functions. Cambridge Univ. Press. Z. · Zbl 0571.60002 [9] KLASS, M. J. 1981. A method of approximating expectations of functions of sum of independent random variables. Ann. Probab. 9 413 428. Z. · Zbl 0463.60023 [10] KLASS, M. J. and ZHANG, C.-H. 1994. On the almost sure minimal growth rate of partial sum maxima. Ann. Probab. 22 1857 1878. Z. · Zbl 0857.60023 [11] KRAKOWIAK, W. and SZULGA, J. 1988. On a p-stable multiple integral II. Probab. Theory Related Fields 78 449 453. Z. · Zbl 0628.60066 [12] LEDOUX, M. and TALAGRAND, M. 1991. Probability in Banach Spaces. Springer, Berlin. Z. · Zbl 0748.60004 [13] MARCINKIEWICZ, J. and ZYGMUND, A. 1937. Sur les fonctions independantes. Fund. Math. 29 60 90. Z. · Zbl 0016.40901 [14] MCCONNELL, T. and TAQQU, M. 1986. Decoupling inequalities for multilinear forms in independent symmetric random variables. Ann. Probab. 14 943 954. Z. · Zbl 0602.60025 [15] PALEY, R. E. A. C. and ZYGMUND, A. 1932. A note on analytic functions on the unit circle. Proc. Cambridge Phil. Soc. 28 266 272. · Zbl 0005.06602 [16] BERKELEY, CALIFORNIA 94720-3860 S-220 07 E-MAIL: klass@stat.berkeley.edu LUND SWEDEN
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