# Convolution property and exponential bounds for symmetric monotone densities

ESAIM: Probability and Statistics (2013)

- Volume: 17, page 605-613
- ISSN: 1292-8100

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topLefèvre, Claude, and Utev, Sergey. "Convolution property and exponential bounds for symmetric monotone densities." ESAIM: Probability and Statistics 17 (2013): 605-613. <http://eudml.org/doc/273631>.

@article{Lefèvre2013,

abstract = {Our first theorem states that the convolution of two symmetric densities which are k-monotone on (0,∞) is again (symmetric) k-monotone provided 0 < k ≤ 1. We then apply this result, together with an extremality approach, to derive sharp moment and exponential bounds for distributions having such shape constrained densities.},

author = {Lefèvre, Claude, Utev, Sergey},

journal = {ESAIM: Probability and Statistics},

keywords = {multiply monotonicity; symmetric densities; unimodality; Wintner’s theorem; Bernstein’s inequality; convolution; -monotone; moment bounds; large deviations},

language = {eng},

pages = {605-613},

publisher = {EDP-Sciences},

title = {Convolution property and exponential bounds for symmetric monotone densities},

url = {http://eudml.org/doc/273631},

volume = {17},

year = {2013},

}

TY - JOUR

AU - Lefèvre, Claude

AU - Utev, Sergey

TI - Convolution property and exponential bounds for symmetric monotone densities

JO - ESAIM: Probability and Statistics

PY - 2013

PB - EDP-Sciences

VL - 17

SP - 605

EP - 613

AB - Our first theorem states that the convolution of two symmetric densities which are k-monotone on (0,∞) is again (symmetric) k-monotone provided 0 < k ≤ 1. We then apply this result, together with an extremality approach, to derive sharp moment and exponential bounds for distributions having such shape constrained densities.

LA - eng

KW - multiply monotonicity; symmetric densities; unimodality; Wintner’s theorem; Bernstein’s inequality; convolution; -monotone; moment bounds; large deviations

UR - http://eudml.org/doc/273631

ER -

## References

top- [1] F. Balabdaoui and J.A. Wellner, Estimation of a k-monotone density: limit distribution theory and the spline connection. Ann. Stat.35 (2007) 2536–2564. Zbl1129.62019MR2382657
- [2] F. Barthe and A. Koldobsky, Extremal slabs in the cube and the Laplace transform. Adv. Math.174 (2003) 89–114. Zbl1037.52006MR1959893
- [3] F. Barthe, F. Gamboa, L. Lozada-Chang and A. Rouault, Generalized Dirichlet distributions on the ball and moments. ALEA – Latin Amer. J. Probab. Math. Stat.7 (2010) 319–340. Zbl1276.30051MR2737561
- [4] E.M.J. Bertin, I. Cuculescu and R. Theodorescu, Unimodality of Probability Measures. Kluwer, Dordrecht (1997). Zbl0876.60001MR1472974
- [5] S. Dharmadhikari and K. Joag-dev, Unimodality, Convexity, and Applications. Academic, San Diego (1988). Zbl0646.62008MR954608
- [6] M.L. Eaton, A note on symmetric Bernoulli random variables. Ann. Math. Stat.41 (1970) 1223–1226. Zbl0203.51805MR268930
- [7] T. Figiel, P. Hitczenko, W.B. Johnson, G. Schechtman and J. Zinn, Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities. Trans. Amer. Math. Soc.349 (1997) 997–1027. Zbl0867.60006MR1390980
- [8] T. Gneiting, Radial positive definite functions generated by Euclid’s hat. J. Multiv. Anal.69 (1999) 88–119. Zbl0937.60007MR1701408
- [9] W. Hoeffding, The extrema of the expected value of a function of independent random variables. Ann. Math. Stat.26 (1955) 268–275. Zbl0064.38105MR70087
- [10] O. Johnson and C. Goldschmidt, Preservation of log-concavity on summation. ESAIM: PS 10 (2005) 206–215. Zbl1185.60016MR2219340
- [11] A.Y. Khintchine, On unimodal distributions. Izv. Nauchno- Issled. Inst. Mat. Mech. Tomsk. Gos. Univ. 2 (1938) 1–7 (in Russian). Zbl0022.35502JFM64.1197.01
- [12] C. Lefèvre and S. Loisel, On multiply monotone distributions, discrete or continuous, with applications. Working paper, ISFA, Université de Lyon 1 (2011). Zbl1293.62028
- [13] C. Lefèvre and S. Utev, Exact norms of a Stein-type operator and associated stochastic orderings. Probab. Theory Relat. Fields127 (2003) 353–366. Zbl1039.60015MR2018920
- [14] P. Lévy, Extensions d’un théorème de D. Dugué et M. Girault. Wahrscheinlichkeitstheorie1 (1962) 159–173. Zbl0108.31502
- [15] A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications. Academic, New York (1979). Zbl1219.26003MR552278
- [16] A.G. Pakes and J. Navarro, Distributional characterizations through scaling relations. Aust. N. Z. J. Stat.49 (2007) 115–135. Zbl1117.62015MR2392366
- [17] I. Pinelis, Toward the best constant factor for the Rademacher-Gaussian tail comparison. ESAIM: PS 11 (2007) 412–426. Zbl1185.60017MR2339301
- [18] I. Podlubny, Fractional Differential Equations. Academic, San Diego (1999). Zbl0924.34008MR1658022
- [19] E. Rio, Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Springer, Berlin (2000). Zbl0944.60008MR2117923
- [20] M. Shaked and J.G. Shanthikumar, Stochatic Orders. Springer, New York (2007). Zbl1111.62016MR2265633
- [21] S.A. Utev, Extremal problems in moment inequalities, in Limit Theorems of Probability Theory of Trudy Inst. Mat., vol. 5. Nauka Sibirsk. Otdel., Novosibirsk. (1985) 56–75 (in Russian). Zbl0579.60018MR821753
- [22] R.E. Williamson, Multiply monotone functions and their Laplace transforms. Duke Math. J.23 (1956) 189–207. Zbl0070.28501MR77581
- [23] A. Wintner, On a class of Fourier transforms. Amer. J. Math.58 (1936) 45–90. Zbl62.0482.06MR1507134JFM62.0482.06

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