A shorter proof of Kanter's Bessel function concentration bound

Probability Theory and Related Fields

Volumn 139, Issue 1-2, 2007, Pages 191-205

  Mattner, Lutz ^a   Roos, Bero ^b  

^a UNIVERSITY OF LÜBECK   (Germany)

^b UNIVERSITY OF HAMBURG   (Germany)

Author keywords

Analytic inequalities; Bernoulli convolution; Concentration function; Modified Bessel function; Poisson binomial distribution; Symmetric three point convolution; Symmetrized Poisson distribution

Indexed keywords

EID: 34250759840     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-006-0043-0     Document Type: Article

References (23)

1. Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation. Clarendon Press, Oxford (1992)
2. Barbour, A.D., Xia, A.: Poisson perturbations. ESAIM Probab. Stat. 3, 131-150 (1999)
3. Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Springer, Berlin Heidelberg New York (1984)
4. Bickel, P.J., van Zwet, W.R.: On a theorem of Hoeffding. In: Chakravarti, I.M. (ed.) Asymptotic Theory of Statistical Tests and Estimation: in Honor of Wassily Hoeffding. Academic, New York, pp. 307-324 (1980)
5. Boland, P.J., Proschan, F.: The reliability of k out of n systems. Ann. Probab. 11, 760-764 (1983)
6. Bondar, J.V.: Comments and complements to inequalities: theory of majorization and its applications by Albert W. Marshall and Ingram Olkin. Linear Algebra Appl. 199, 115-130 (1994)
7. Bretagnolle, J.: Sur l'inégalité de concentration de Doeblin-Lévy, Rogozin-Kesten. In: Parametric and semiparametric models with applications to reliability, survival analysis, and quality of life, Stat. Ind. Technol., Birkhäuser, Boston, pp. 533-551 (2004)
8. Čekanavičius, V., Roos, B.: An expansion in the exponent for compound binomial approximations. Lith. Math. J. 46, 54-91 (2006)
9. Dharmadhikari, S., Joag-Dev, K.: Unimodality, Convexity, and Applications. Academic, Boston (1988)
10. Gleser, L.: On the distribution of the number of successes in independent trials. Ann. Probab. 3, 182-188 (1975)
11. Hoeffding, W.: On the distribution of the number of successes in independent trials. Ann. Math. Stat. 27, 713-721 (1956).
12. Also in: Fisher, N.I., Sen, P.K. (eds.) The Collected Works of Wassily Hoeffding. Springer, Berlin Heidelberg New York, 1994
13. Kanter, M.: Probability inequalities for convex sets and multidimensional concentration functions. J. Multivariate Anal. 6, 222-236 (1976)
14. Karlin, S., Novikoff, A.: Generalized convex inequalities. Pac. J. Math. 13, 1251-1279 (1963)
15. Le Cam, L.: Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York (1986)
16. Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications. Academic, New York (1979)
17. Mattner, L.: Lower bounds for tails of sums of independent symmetric random variables. Preprint, http://arxiv.org/abs/math.PR/0609200 (2006)
18. Merkle, M., Petrović, L.: Inequalities for sums independent geometrical random variables. Aequ. Math. 54, 173-180 (1997)
19. Olver, F.W.J.: Asymptotics and Special Functions. AK Peters, Wellesley (1997)
20. Petrov, V.V.: Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Clarendon Press, Oxford (1995)
21. Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis II. 4. Auflage, Springer, Berlin Heidelberg New York (1971)
22. Rogozin, B. A.: Inequalities for concentration of a decomposition. Theory Probab. Appl. 38, 556-562 (1993)
23. Roos, B.: On Hipp's compound Poisson approximations via concentration functions. Bernoulli 11, 533-557 (2005)