# Preservation of log-concavity on summation

Oliver Johnson; Christina Goldschmidt

ESAIM: Probability and Statistics (2006)

- Volume: 10, page 206-215
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topJohnson, Oliver, and Goldschmidt, Christina. "Preservation of log-concavity on summation." ESAIM: Probability and Statistics 10 (2006): 206-215. <http://eudml.org/doc/249749>.

@article{Johnson2006,

abstract = {
We extend Hoggar's theorem that the sum of two independent
discrete-valued log-concave random variables is itself log-concave. We
introduce conditions under which the result still holds for dependent
variables. We argue that these conditions are natural by giving some
applications. Firstly, we use our main theorem to give simple proofs
of the log-concavity of the Stirling numbers of the second kind and of
the Eulerian numbers.
Secondly, we prove results concerning the log-concavity
of the sum of independent (not necessarily log-concave) random
variables.
},

author = {Johnson, Oliver, Goldschmidt, Christina},

journal = {ESAIM: Probability and Statistics},

keywords = {Log-concavity; convolution; dependent random
variables; Stirling numbers; Eulerian numbers.; log-concavity; dependent random variables; Eulerian numbers},

language = {eng},

month = {5},

pages = {206-215},

publisher = {EDP Sciences},

title = {Preservation of log-concavity on summation},

url = {http://eudml.org/doc/249749},

volume = {10},

year = {2006},

}

TY - JOUR

AU - Johnson, Oliver

AU - Goldschmidt, Christina

TI - Preservation of log-concavity on summation

JO - ESAIM: Probability and Statistics

DA - 2006/5//

PB - EDP Sciences

VL - 10

SP - 206

EP - 215

AB -
We extend Hoggar's theorem that the sum of two independent
discrete-valued log-concave random variables is itself log-concave. We
introduce conditions under which the result still holds for dependent
variables. We argue that these conditions are natural by giving some
applications. Firstly, we use our main theorem to give simple proofs
of the log-concavity of the Stirling numbers of the second kind and of
the Eulerian numbers.
Secondly, we prove results concerning the log-concavity
of the sum of independent (not necessarily log-concave) random
variables.

LA - eng

KW - Log-concavity; convolution; dependent random
variables; Stirling numbers; Eulerian numbers.; log-concavity; dependent random variables; Eulerian numbers

UR - http://eudml.org/doc/249749

ER -

## References

top- T. Bergstrom and M. Bagnoli, Log-concave probability and its applications. Econom. Theory26 (2005) 445–469.
- B. Biais, D. Martimort and J.-C. Rochet, Competing mechanisms in a common value environment. Econometrica68 (2000) 799–837.
- M. Bóna and R. Ehrenborg, A combinatorial proof of the log-concavity of the numbers of permutations with k runs. J. Combin. Theory Ser. A90 (2000) 293–303.
- F. Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc.81 (1989) viii+106.
- F. Brenti, Expansions of chromatic polynomials and log-concavity. Trans. Amer. Math. Soc.332 (1992) 729–756.
- F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update in Jerusalem combinatorics '93, Amer. Math. Soc., Providence, RI, Contemp. Math.178 (1994) 71–89.
- H. Davenport and G. Pólya, On the product of two power series. Canadian J. Math.1 (1949) 1–5.
- V. Gasharov, On the Neggers-Stanley conjecture and the Eulerian polynomials. J. Combin. Theory Ser. A82 (1998) 134–146.
- S.G. Hoggar, Chromatic polynomials and logarithmic concavity. J. Combin. Theory Ser. B16 (1974) 248–254.
- K. Joag-Dev and F. Proschan, Negative association of random variables with applications. Ann. Statist.11 (1983) 286–295.
- E.H. Lieb, Concavity properties and a generating function for Stirling numbers. J. Combin. Theory5 (1968) 203–206.
- E.J. Miravete, Preserving log-concavity under convolution: Comment. Econometrica70 (2002) 1253–1254.
- C.P. Niculescu, A new look at Newton's inequalities. JIPAM. J. Inequal. Pure Appl. Math.1 (2000) Issue 2, Article 17; see also . URIhttp://jipam.vu.edu.au/
- R.C. Read, An introduction to chromatic polynomials. J. Combin. Theory4 (1968) 52–71.
- B.E. Sagan, Inductive and injective proofs of log concavity results. Discrete Math.68 (1988) 281–292.
- B.E. Sagan, Inductive proofs of q-log concavity. Discrete Math.99 (1992) 289–306.
- R.P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, in Graph theory and its applications: East and West (Jinan, 1986), Ann. New York Acad. Sci., New York Acad. Sci., New York 576 (1989) 500–535.
- Y. Wang, Linear transformations preserving log-concavity. Linear Algebra Appl.359 (2003) 161–167.
- Y. Wang and Y.-N. Yeh, Log-concavity and LC-positivity. Available at arXiv:math. (2005). To appear in J. Combin. Theory Ser A. URICO/0504164
- Y. Wang and Y.-N. Yeh, Polynomials with real zeros and Pólya frequency sequences. J. Combin. Theory Ser. A109 (2005) 63–74.
- D.J.A. Welsh, Matroid theory, L.M.S. Monographs, No. 8. Academic Press, London (1976).
- H.S. Wilf, Generatingfunctionology. Academic Press Inc., Boston, MA, second edition (1994).

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.