PlanetMath: subadditivity

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subadditivity

(Definition)

A sequence $\{a_n\}_{n=1}^\infty$ is called subadditive if it satisfies the inequality

$\displaystyle a_{n+m}\leq a_n+a_m$ for all

and

$\displaystyle .$

(1)

The major reason for use of subadditive sequences is the following lemma due to Fekete.

Lemma 1 ([1]) For every subadditive sequence $\{a_n\}_{n=1}^\infty$ the limit $\lim a_n/n$ exists and is equal to $\inf a_n/n$ .

Similarly, a function is subadditive if

$\displaystyle f(x+y)\leq f(x)+f(y)$ for all

and

$\displaystyle .$

The analogue of Fekete lemma holds for subadditive functions as well.

There are extensions of Fekete lemma that do not require (1) to hold for all and . 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.
Zbl 0338.00001.
2: Michael J. Steele.
Probability theory and combinatorial optimization, volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics.
SIAM, 1997.
Zbl 0916.90233.

"subadditivity" is owned by bbukh.

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