Abstract
We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide sharp upper bounds for the sum of modified Bessel functions I0(x) + I1(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent random variables due to Čekanavičius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).
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Barbour A.D., Holst L., Janson S. (1992) Poisson Approximation. Clarendon Press, Oxford
Barbour A.D., Xia A. (1999) Poisson perturbations. ESAIM Probab. Stat. 3, 131–150
Berg C., Christensen J.P.R., Ressel P. (1984) Harmonic Analysis on Semigroups. Springer, Berlin Heidelberg New York
Bickel P.J., van Zwet W.R. (1980) On a theorem of Hoeffding. In: Chakravarti I.M. (eds). Asymptotic Theory of Statistical Tests and Estimation: in Honor of Wassily Hoeffding. Academic, New York, pp. 307–324
Boland P.J., Proschan F. (1983) The reliability of k out of n systems. Ann. Probab. 11, 760–764
Bondar J.V. (1994) Comments and complements to inequalities: theory of majorization and its applications by Albert W. Marshall and Ingram Olkin. Linear Algebra Appl. 199, 115–130
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)
Čekanavičius V., Roos B. (2006) An expansion in the exponent for compound binomial approximations. Lith. Math. J. 46, 54–91
Dharmadhikari S., Joag-Dev K. (1988) Unimodality, Convexity, and Applications. Academic, Boston
Gleser L. (1975) On the distribution of the number of successes in independent trials. Ann. Probab. 3, 182–188
Hoeffding, W.: On the distribution of the number of successes in independent trials. Ann. Math. Stat. 27, 713–721 (1956). Also in: Fisher, N.I., Sen, P.K. (eds.) The Collected Works of Wassily Hoeffding. Springer, Berlin Heidelberg New York, 1994
Kanter M. (1976) Probability inequalities for convex sets and multidimensional concentration functions. J. Multivariate Anal. 6, 222–236
Karlin S., Novikoff A. (1963) Generalized convex inequalities. Pac. J. Math. 13, 1251–1279
Le Cam L. (1986) Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York
Marshall A.W., Olkin I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic, New York
Mattner, L.: Lower bounds for tails of sums of independent symmetric random variables. Preprint, http://arxiv.org/abs/math.PR/0609200 (2006)
Merkle M., Petrović L. (1997) Inequalities for sums independent geometrical random variables. Aequ. Math. 54, 173–180
Olver F.W.J. (1997) Asymptotics and Special Functions. AK Peters, Wellesley
Petrov V.V. (1995) Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Clarendon Press, Oxford
Pólya G., Szegö G. (1971) Aufgaben und Lehrsätze aus der Analysis II. 4. Auflage. Springer, Berlin Heidelberg New York
Rogozin B.A. (1993) Inequalities for concentration of a decomposition. Theory Probab. Appl. 38, 556–562
Roos B. (2005) On Hipp’s compound Poisson approximations via concentration functions. Bernoulli 11, 533–557
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Mattner, L., Roos, B. A shorter proof of Kanter’s Bessel function concentration bound. Probab. Theory Relat. Fields 139, 191–205 (2007). https://doi.org/10.1007/s00440-006-0043-0
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DOI: https://doi.org/10.1007/s00440-006-0043-0
Keywords
- Analytic inequalities
- Bernoulli convolution
- Modified Bessel function
- Concentration function
- Poisson binomial distribution
- Symmetric three point convolution
- Symmetrized Poisson distribution