## The Kadec-Pełczyński theorem in $L^p$, $1\le p<2$

HTML articles powered by AMS MathViewer

- by I. Berkes and R. Tichy PDF
- Proc. Amer. Math. Soc.
**144**(2016), 2053-2066 Request permission

## Abstract:

By a classical result of Kadec and Pełczyński (1962), every normalized weakly null sequence in $L^p$, $p>2$, contains a subsequence equivalent to the unit vector basis of $\ell ^2$ or to the unit vector basis of $\ell ^p$. In this paper we investigate the case $1\le p<2$ and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczyński theorem is that the limit random measure $\mu$ of the sequence satisfies $\int _{\mathbb {R}} x^2 d\mu (x)\in L^{p/2}$.## References

- D. J. Aldous,
*Limit theorems for subsequences of arbitrarily-dependent sequences of random variables*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**40**(1977), no. 1, 59–82. MR**455090**, DOI 10.1007/BF00535707 - I. Berkes,
*On almost symmetric sequences in $L_p$*, Acta Math. Hungar.**54**(1989), no. 3-4, 269–278. MR**1029089**, DOI 10.1007/BF01952057 - István Berkes and Erika Péter,
*Exchangeable random variables and the subsequence principle*, Probab. Theory Related Fields**73**(1986), no. 3, 395–413. MR**859840**, DOI 10.1007/BF00776240 - Patrick Billingsley,
*Convergence of probability measures*, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR**1700749**, DOI 10.1002/9780470316962 - István Berkes and Haskell P. Rosenthal,
*Almost exchangeable sequences of random variables*, Z. Wahrsch. Verw. Gebiete**70**(1985), no. 4, 473–507. MR**807333**, DOI 10.1007/BF00531863 - D. Dacunha-Castelle,
*Indiscernability and exchangeability in $L^{p}$-spaces*, Proceedings of the Seminar on Random Series, Convex Sets and Geometry of Banach Spaces (Mat. Inst., Aarhus Univ., Aarhus, 1974; dedicated to the memory of E. Asplund), Various Publications Series, No. 24, Mat. Inst., Aarhus Univ., Aarhus, 1975, pp. 50–56. MR**0385948** - C. G. Esseen,
*On the concentration function of a sum of independent random variables*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**9**(1968), 290–308. MR**231419**, DOI 10.1007/BF00531753 - William Feller,
*An introduction to probability theory and its applications. Vol. II.*, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0270403** - Sylvie Guerre,
*Types et suites symétriques dans $L^p,\;1\leq p<+\infty ,\;p\not = 2$*, Israel J. Math.**53**(1986), no. 2, 191–208 (French, with English summary). MR**845871**, DOI 10.1007/BF02772858 - S. Guerre and Y. Raynaud,
*On sequences with no almost symmetric subsequence*, Texas Functional Analysis Seminar 1985–1986 (Austin, TX, 1985–1986) Longhorn Notes, Univ. Texas, Austin, TX, 1986, pp. 83–93. MR**1017045** - M. I. Kadec and A. Pełczyński,
*Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$*, Studia Math.**21**(1961/62), 161–176. MR**152879**, DOI 10.4064/sm-21-2-161-176 - P. Lévy,
*Théorie de l’addition des variables aléatoires*, Gauthier-Villars, 1937. - J. Marczinkiewicz and A. Zygmund,
*Quelques théorèmes sur les fonctions indépendantes*, Studia Math.**7**(1938), 104–120. - R. Ranga Rao,
*Relations between weak and uniform convergence of measures with applications*, Ann. Math. Statist.**33**(1962), 659–680. MR**137809**, DOI 10.1214/aoms/1177704588 - Alfréd Rényi,
*On stable sequences of events*, Sankhyā Ser. A**25**(1963), 293 302. MR**170385**

## Additional Information

**I. Berkes**- Affiliation: Institute of Statistics, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria
- MR Author ID: 35400
- Email: berkes@tugraz.at
**R. Tichy**- Affiliation: Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
- MR Author ID: 172525
- Email: tichy@tugraz.at
- Received by editor(s): September 8, 2014
- Received by editor(s) in revised form: May 27, 2015
- Published electronically: September 15, 2015
- Additional Notes: The research of the first author was supported by FWF grant P24302-N18 and OTKA grant K 108615.

The research of the second author was supported by FWF grant SFB F5510. - Communicated by: Thomas Schlumprecht
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 2053-2066 - MSC (2010): Primary 46B09, 46B25
- DOI: https://doi.org/10.1090/proc/12872
- MathSciNet review: 3460166