superadditivity
A sequence ${\{{a}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is called superadditive if it satisfies the inequality^{}
$${a}_{n+m}\ge {a}_{n}+{a}_{m}\mathit{\hspace{1em}\hspace{1em}}\text{for all}n\text{and}m.$$ | (1) |
The major reason for use of superadditive sequences is the following lemma due to Fekete.
Lemma ([1]).
For every superadditive sequence ${\mathrm{\{}{a}_{n}\mathrm{\}}}_{n\mathrm{=}\mathrm{1}}^{\mathrm{\infty}}$ the limit $\mathrm{lim}\mathit{}{a}_{n}\mathrm{/}n$ exists and is equal to $\mathrm{sup}\mathit{}{a}_{n}\mathrm{/}n$.
Similarly, a function $f(x)$ is superadditive if
$$f(x+y)\ge f(x)+f(y)\mathit{\hspace{1em}\hspace{1em}}\text{for all}x\text{and}y.$$ |
The analogue of Fekete lemma holds for superadditive functions as well.
There are extensions^{} of Fekete lemma that do not require (1) to hold for all $m$ and $n$. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete lemma if some kind of both super- and subadditivity is present. A good exposition of this topic may be found in [2].
References
- 1 György Polya and Gábor Szegö. Problems and theorems in analysis^{}, volume 1. 1976. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0338.00001Zbl 0338.00001.
- 2 Michael J. Steele. Probability theory and combinatorial optimization, volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1997. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0916.90233Zbl 0916.90233.
Title | superadditivity |
---|---|
Canonical name | Superadditivity |
Date of creation | 2013-03-22 13:52:25 |
Last modified on | 2013-03-22 13:52:25 |
Owner | bbukh (348) |
Last modified by | bbukh (348) |
Numerical id | 10 |
Author | bbukh (348) |
Entry type | Definition |
Classification | msc 39B62 |
Synonym | superadditive |
Related topic | Subadditivity |