×

zbMATH — the first resource for mathematics

On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables. (English) Zbl 1228.60039
Let \(\{X_i: i\geq 1\}\) be a sequence of nonnegative, superadditive, dependent and uniformly bounded random variables, and consider the weighted quadratic form \[ Q_n:= \sum_{1\leq i< j\leq n} a_{ij}X,\quad n\geq 2, \] where \(\{a_{ij}: 1\leq i<j\leq n\}\) is an array of real numbers. Some probability inequalities of Kolmogorov type are obtained for weighted quadratic forms. A number of examples are also presented to demonstrate the applicability of these inequalities.

MSC:
60F15 Strong limit theorems
60E15 Inequalities; stochastic orderings
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Basu, A.K., Probability inequalities for sums of dependent random vector and applications, Sankkya: the Indian J. statist., 47, 333-349, (1985), Series A, Pt. 3 · Zbl 0591.60024
[2] Christofides, T.C., Probability inequalities with exponential bounds for \(U\)-statistics, Statist. probab. lett., 78, 3294-3297, (1991)
[3] Christofides, T.C., A Kolmogorov inequality for \(U\)-statistics based on Bernoulli kernels, J. statist. plann. inference, 21, 357-362, (1994) · Zbl 0810.60013
[4] Chung, F.; Lu, L., Concentration inequalities and martingale inequalities: a survey, Internet math., 3, 1, 79-127, (2006) · Zbl 1111.60010
[5] Cuzich, J.; Gine, E.; Zinn, J., Laws of large numbers for quadratic forms, maxima of products and truncated sums of i.i.d. random variables, Ann. probab., 23, 292-333, (1995) · Zbl 0833.60030
[6] Eghbal, N.; Amini, M.; Bozorgnia, A., Some maximal inequalities for quadratic forms of negative superadditive dependence random variables, Statist. probab. lett., 80, 587-591, (2010) · Zbl 1187.60020
[7] Hoeffding, W., A class of statistics with asymptotically normal distributions, Ann. math. statist., 19, 293-325, (1948) · Zbl 0032.04101
[8] Hoeffding, W., Probability inequalities for sums of bounded random variables, J. amer. stat. assoc., 58, 13-30, (1963) · Zbl 0127.10602
[9] Hu, T., Negatively superadditive dependence of random variables with applications, Chinese J. appl. probab. statist., 16, 133-144, (2000) · Zbl 1050.60502
[10] Kemperman, J.H.B., On the FKG-inequalities for measures on a partially ordered space, Proc. akad. wetenschappen, ser. A, 80, 313-331, (1977) · Zbl 0384.28012
[11] Mari, D.D.; Kotz, S., Correlation and dependence, (2001), Imperial College Press London · Zbl 0977.62004
[12] Mavrikiou, P.M., Kolmogorov inequalities for the partial sum of independent Bernoulli random variables, Statist. probab. lett., 77, 1117-1122, (2007) · Zbl 1120.60014
[13] Mavrikiou, P.M., A Kolmogorov inequality for weighted \(U\)-statistics, Statist. probab. lett., 78, 3294-3297, (2008) · Zbl 05380119
[14] Serfling, R.J., Approximation theorems of mathematical statistics, (1980), Wiley New York · Zbl 0423.60030
[15] Turner, D.W.; Young, D.M.; Seaman, J.W., A Kolmogorov inequality for the sum of independent Bernoulli random variables with unequal means, Statist. probab. lett., 23, 243-245, (1995) · Zbl 0824.60023
[16] Varberg, D.E., Convergence of quadratic forms in independent random variables, Ann. math. statist., 37, 567-575, (1966) · Zbl 0143.20602
[17] Whittle, P., Bounds for the moments of linear and quadratic forms in independent variables, Theory probab. appl., 5, 303-305, (1960)
[18] Whittle, P., On the convergence to normality of quadratic forms in independent variables, Theory probab. appl., 9, 103-109, (1964)
[19] Young, D.M.; Seaman, W.J.; Marco, R.V., A note on a Kolmogorov inequality, Statist. probab. lett., 5, 217-218, (1987) · Zbl 0616.60021
[20] Zhang, C.H., Strong law of large numbers for sums of products, Ann. probab., 24, 1589-1615, (1996) · Zbl 0868.60024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.