Moment inequalities for sums of certain independent symmetric random variables

Hitczenko, P.; Montgomery-Smith, S.; Oleszkiewicz, K.

Hitczenko, P. ; Montgomery-Smith, S. ; Oleszkiewicz, K.

Studia Mathematica, Tome 122 (1997), p. 15-42 / Harvested from The Polish Digital Mathematics Library

Résumé

This paper gives upper and lower bounds for moments of sums of independent random variables $(X_{k})$ which satisfy the condition ${P (| X |}_{k} \geq t) = e x p (- N_{k} (t))$ , where $N_{k}$ are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which $N (t) = {| t |}^{r}$ for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.

Publié le : 1997-01-01

Zbl 0967.60018

EUDML-ID : urn:eudml:doc:216377

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     author = {P. Hitczenko and S. Montgomery-Smith and K. Oleszkiewicz},
     title = {Moment inequalities for sums of certain independent symmetric random variables},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {15-42},
     zbl = {0967.60018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv123i1p15bwm}
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Hitczenko, P.; Montgomery-Smith, S.; Oleszkiewicz, K. Moment inequalities for sums of certain independent symmetric random variables. Studia Mathematica, Tome 122 (1997) pp. 15-42. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv123i1p15bwm/

Bibliographie

[00000] T. Figiel, P. Hitczenko, W. B. Johnson, G. Schechtman, and J. Zinn (1994), Extremal properties of Rademacher functions with applications to Khintchine and Rosenthal inequalities, Trans. Amer. Math. Soc., to appear. | Zbl 0867.60006

[00001] T. Figiel, T. Iwaniec and A. Pełczyński (1984), Computing norms and critical exponents of some operators in $L_{p}$ -spaces, Studia Math. 79, 227-274. | Zbl 0599.46035

[00002] E. D. Gluskin and S. Kwapień (1995), Tail and moment estimates for sums of independent random variables with logarithmically concave tails, ibid. 114, 303-309. | Zbl 0834.60050

[00003] U. Haagerup (1982), Best constants in the Khintchine's inequality, ibid. 70, 231-283. | Zbl 0501.46015

[00004] M. G. Hahn and M. J. Klass (1995), Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential bound, preprint.

[00005] P. Hitczenko (1993), Domination inequality for martingale transforms of Rademacher sequence, Israel J. Math. 84, 161-178. | Zbl 0781.60037

[00006] P. Hitczenko (1994), On a domination of sums of random variables by sums of conditionally independent ones, Ann. Probab. 22, 453-468. | Zbl 0809.60022

[00007] P. Hitczenko and S. Kwapień (1994), On the Rademacher series, in: Probability in Banach Spaces, 9, Sandbjerg 1993, J. Hoffmann-Jørgensen, J. Kuelbs and M. B. Marcus (eds.), Birkhäuser, Boston, 1994, 31-36. | Zbl 0822.60013

[00008] N. L. Johnson and S. Kotz (1970), Continuous Univariate Distributions, Houghton-Mifflin, New York. | Zbl 0213.21101

[00009] W. B. Johnson, G. Schechtman and J. Zinn (1983), Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Probab. 13, 234-253. | Zbl 0564.60020

[00010] M. Klass (1976), Precision bounds for the relative error in the approximation of $E | S_{n} |$ and extensions, ibid. 8, 350-367. | Zbl 0428.60058

[00011] S. Kwapień and J. Szulga (1991), Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces, ibid. 19, 1-8. | Zbl 0718.60044

[00012] S. Kwapień and W. A. Woyczyński (1992), Random Series and Stochastic Integrals. Single and Multiple, Birkhäuser, Boston. | Zbl 0751.60035

[00013] M. Ledoux and M. Talagrand (1991), Probability in Banach Spaces, Springer, Berlin. | Zbl 0748.60004

[00014] A. W. Marshall and I. Olkin (1979), Inequalities: Theory of Majorization and Its Applications, Academic Press, New York. | Zbl 0437.26007

[00015] S. J. Montgomery-Smith (1990), The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109, 517-522. | Zbl 0696.60013

[00016] S. J. Montgomery-Smith (1992), Comparison of Orlicz-Lorentz spaces, Studia Math. 103, 161-189. | Zbl 0814.46023

[00017] I. Pinelis (1994), Extremal probabilistic problems and Hotelling’s $T^{2}$ test under a symmetry condition, Ann. Statist. 22, 357-368. | Zbl 0812.62065

[00018] I. Pinelis (1994), Optimum bounds for the distributions of martingales in Banach spaces, Ann. Probab. 22, 1679-1706. | Zbl 0836.60015

[00019] H. P. Rosenthal (1970), On the subspaces of $L_{p}$ (p>2) spanned by sequences of independent random variables, Israel J. Math. 8, 273-303. | Zbl 0213.19303

[00020] G. Schechtman and J. Zinn (1990), On the volume of the intersection of two $L_{p}^{n}$ balls, Proc. Amer. Math. Soc. 110, 217-224. | Zbl 0704.60017

[00021] M. Talagrand (1989), Isoperimetry and integrability of the sum of independent Banach-space valued random variables, Ann. Probab. 17, 1546-1570. | Zbl 0692.60016

[00022] S. A. Utev (1985), Extermal problems in moment inequalities, in: Limit Theorems in Probability Theory, Trudy Inst. Mat., Novosibirsk, 1985, 56-75 (in Russian).