A probabilistic version of Rosenthal’s inequality

Astashkin, S.; Tikhomirov, K.

doi:10.1090/S0002-9939-2013-11713-2

A probabilistic version of Rosenthal’s inequality
HTML articles powered by AMS MathViewer

by S. V. Astashkin and K. E. Tikhomirov PDF

Proc. Amer. Math. Soc. 141 (2013), 3539-3547 Request permission

Abstract:

We prove some probabilistic relations between sums of independent random variables and the corresponding disjoint sums, which strengthen the well-known Rosenthal inequality and its generalizations. As a consequence we extend the inequalities proved earlier by Montgomery-Smith and Junge for rearrangement invariant spaces to the quasi-normed case.

References

William B. Johnson and G. Schechtman, Sums of independent random variables in rearrangement invariant function spaces, Ann. Probab. 17 (1989), no. 2, 789–808. MR 985390
Haskell P. Rosenthal, On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303. MR 271721, DOI 10.1007/BF02771562
S. V. Astashkin and F. A. Sukochev, Series of independent random variables in rearrangement invariant spaces: an operator approach, Israel J. Math. 145 (2005), 125–156. MR 2154724, DOI 10.1007/BF02786688
S. V. Astashkin and F. A. Sukochev, Comparison of sums of independent and disjoint functions in symmetric spaces, Mat. Zametki 76 (2004), no. 4, 483–489 (Russian, with Russian summary); English transl., Math. Notes 76 (2004), no. 3-4, 449–454. MR 2112064, DOI 10.1023/B:MATN.0000043474.00734.ec
S. V. Astashkin and F. A. Sukochev, Series of zero-mean independent functions in symmetric spaces with the Kruglov property, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 345 (2007), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 34, 25–50, 140 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 148 (2008), no. 6, 795–809. MR 2432174, DOI 10.1007/s10958-008-0026-z
PawełHitczenko and Stephen Montgomery-Smith, Measuring the magnitude of sums of independent random variables, Ann. Probab. 29 (2001), no. 1, 447–466. MR 1825159, DOI 10.1214/aop/1008956339
E. M. Nikišin, Resonance theorems and superlinear operators, Uspehi Mat. Nauk 25 (1970), no. 6(156), 129–191 (Russian). MR 0296584
Stephen Montgomery-Smith, Rearrangement invariant norms of symmetric sequence norms of independent sequences of random variables, Israel J. Math. 131 (2002), 51–60. MR 1942301, DOI 10.1007/BF02785850
Marius Junge, The optimal order for the $p$-th moment of sums of independent random variables with respect to symmetric norms and related combinatorial estimates, Positivity 10 (2006), no. 2, 201–230. MR 2237498, DOI 10.1007/s11117-005-0008-z
S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9