Subadditivity

In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions.

Definitions

A subadditive function is a function ${\displaystyle f\colon A\to B}$, having a domain A and an ordered codomain B that are both closed under addition, with the following property:

${\displaystyle \forall x,y\in A,f(x+y)\leq f(x)+f(y).}$

An example is the square root function, having the non-negative real numbers as domain and codomain, since ${\displaystyle \forall x,y\geq 0}$ we have:

${\displaystyle {\sqrt {x+y}}\leq {\sqrt {x}}+{\sqrt {y}}.}$

A sequence ${\displaystyle \left\{a_{n}\right\},n\geq 1}$, is called subadditive if it satisfies the inequality

${\displaystyle (1)\qquad a_{n+m}\leq a_{n}+a_{m}}$

for all m and n. This is a special case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers.

Properties

Sequences

A useful result pertaining to subadditive sequences is the following lemma due to Michael Fekete.[1]

Fekete's Subadditive Lemma: For every subadditive sequence ${\displaystyle {\left\{a_{n}\right\}}_{n=1}^{\infty }}$, the limit ${\displaystyle \displaystyle \lim _{n\to \infty }{\frac {a_{n}}{n}}}$ exists and is equal to the infimum ${\displaystyle \inf {\frac {a_{n}}{n}}}$. (The limit may be ${\displaystyle -\infty }$.)

The analogue of Fekete's lemma holds for superadditive sequences as well, that is: ${\displaystyle a_{n+m}\geq a_{n}+a_{m}.}$ (The limit then may be positive infinity: consider the sequence ${\displaystyle a_{n}=\log n!}$.)

There are extensions of Fekete's lemma that do not require the inequality (1) to hold for all m and n, but only for m and n such that ${\displaystyle {\frac {1}{2}}\leq {\frac {m}{n}}\leq 2.}$ Moreover, the condition ${\displaystyle a_{n+m}\leq a_{n}+a_{m}}$ may be weakened as follows: ${\displaystyle a_{n+m}\leq a_{n}+a_{m}+\phi (n+m)}$ provided that ${\displaystyle \phi }$ is an increasing function such that the integral ${\displaystyle \int \phi (t)t^{-2}\,dt}$ converges (near the infinity).[2]

There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present.[3][4]

Besides, analogues of Fekete's lemma have been proved for subadditive real maps (with additional assumptions) from finite subsets of an amenable group [5] [6] ,[7] and further, of a cancellative left-amenable semigroup.[8]

Functions

Theorem:[9] For every measurable subadditive function ${\displaystyle f:(0,\infty )\to \mathbb {R} ,}$ the limit ${\displaystyle \lim _{t\to \infty }{\frac {f(t)}{t}}}$ exists and is equal to ${\displaystyle \inf _{t>0}{\frac {f(t)}{t}}.}$ (The limit may be ${\displaystyle -\infty .}$)

If f is a subadditive function, and if 0 is in its domain, then f(0) ≥ 0. To see this, take the inequality at the top. ${\displaystyle f(x)\geq f(x+y)-f(y)}$. Hence ${\displaystyle f(0)\geq f(0+y)-f(y)=0}$

A concave function ${\displaystyle f:[0,\infty )\to [0,\infty )}$ with ${\displaystyle f(0)\geq 0}$ is also subadditive. To see this, one first observes that ${\displaystyle f(x)\geq \textstyle {\frac {y}{x+y}}f(0)+\textstyle {\frac {x}{x+y}}f(x+y)}$. Then looking at the sum of this bound for ${\displaystyle f(x)}$ and ${\displaystyle f(y)}$, will finally verify that f is subadditive.[10]

The negative of a subadditive function is superadditive.

Examples in various domains

Entropy

Entropy plays a fundamental role in information theory and statistical physics, as well as in quantum mechanics in a generalized formulation due to von Neumann. Entropy appears always as a subadditive quantity in all of its formulations, meaning the entropy of a supersystem or a set union of random variables is always less or equal than the sum of the entropies of its individual components. Additionally, entropy in physics satisfies several more strict inequalities such as the Strong Subadditivity of Entropy in classical statistical mechanics and its quantum analog.

Economics

Subadditivity is an essential property of some particular cost functions. It is, generally, a necessary and sufficient condition for the verification of a natural monopoly. It implies that production from only one firm is socially less expensive (in terms of average costs) than production of a fraction of the original quantity by an equal number of firms.

Economies of scale are represented by subadditive average cost functions.

Except in the case of complementary goods, the price of goods (as a function of quantity) must be subadditive. Otherwise, if the sum of the cost of two items is cheaper than the cost of the bundle of two of them together, then nobody would ever buy the bundle, effectively causing the price of the bundle to "become" the sum of the prices of the two separate items. Thus proving that it is not a sufficient condition for a natural monopoly; since the unit of exchange may not be the actual cost of an item. This situation is familiar to everyone in the political arena where some minority asserts that the loss of some particular freedom at some particular level of government means that many governments are better; whereas the majority assert that there is some other correct unit of cost.

Finance

Subadditivity is one of the desirable properties of coherent risk measures in risk management.[11] The economic intuition behind risk measure subadditivity is that a portfolio risk exposure should, at worst, simply equal the sum of the risk exposures of the individual positions that compose the portfolio. In any other case the effects of diversification would result in a portfolio exposure that is lower than the sum of the individual risk exposures. The lack of subadditivity is one of the main critiques of VaR models which do not rely on the assumption of normality of risk factors. The Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio ${\displaystyle V}$ at the confidence level ${\displaystyle 1-p}$ is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss,

${\displaystyle {\text{VaR}}_{p}\equiv z_{p}\sigma _{\Delta V}=z_{p}{\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+2\rho _{xy}\sigma _{x}\sigma _{y}}}}$

where ${\displaystyle z_{p}}$ is the inverse of the normal cumulative distribution function at probability level ${\displaystyle p}$, ${\displaystyle \sigma _{x}^{2},\sigma _{y}^{2}}$ are the individual positions returns variances and ${\displaystyle \rho _{xy}}$ is the linear correlation measure between the two individual positions returns. Since variance is always positive,

${\displaystyle {\sqrt {\sigma _{x}^{2}+\sigma _{y}^{2}+2\rho _{xy}\sigma _{x}\sigma _{y}}}\leq \sigma _{x}+\sigma _{y}}$

Thus the Gaussian VaR is subadditive for any value of ${\displaystyle \rho _{xy}\in [-1,1]}$ and, in particular, it equals the sum of the individual risk exposures when ${\displaystyle \rho _{xy}=1}$ which is the case of no diversification effects on portfolio risk.

Thermodynamics

Subadditivity occurs in the thermodynamic properties of non-ideal solutions and mixtures like the excess molar volume and heat of mixing or excess enthalpy.

Combinatorics on words

A factorial language ${\displaystyle L}$ is one where if a word is in ${\displaystyle L}$, then all factors of that word are also in ${\displaystyle L}$. In combinatorics on words, a common problem is to determine the number ${\displaystyle A(n)}$ of length-${\displaystyle n}$ words in a factorial language. Clearly ${\displaystyle A(m+n)\leq A(m)A(n)}$, so ${\displaystyle \log A(n)}$ is subadditive, and hence Fekete's lemma can be used to estimate the growth of ${\displaystyle A(n)}$. [12]

See also

• Triangle inequality
• Superadditivity
• Choquet integral
• Apparent molar property

Notes

1. Fekete, M. (1923). "Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten". Mathematische Zeitschrift. 17 (1): 228–249. doi:10.1007/BF01504345.
2. de Bruijn, N.G.; Erdös, P. (1952). "Some linear and some quadratic recursion formulas. II". Nederl. Akad. Wetensch. Proc. Ser. A. 55: 152–163. doi:10.1016/S1385-7258(52)50021-0. (The same as Indagationes Math. 14.) See also Steele 1997, Theorem 1.9.2.
3. Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997). ISBN 0-89871-380-3.
4. Michael J. Steele (2011). CBMS Lectures on Probability Theory and Combinatorial Optimization. University of Cambridge.
5. Lindenstrauss, Elon; Weiss, Benjamin (2000). "Mean topological dimension". Israel Journal of Mathematics. 115 (1): 1–24. CiteSeerX 10.1.1.30.3552. doi:10.1007/BF02810577. ISSN 0021-2172. Theorem 6.1
6. Ornstein, Donald S.; Weiss, Benjamin (1987). "Entropy and isomorphism theorems for actions of amenable groups". Journal d'Analyse Mathématique. 48 (1): 1–141. doi:10.1007/BF02790325. ISSN 0021-7670.
7. Gromov, Misha (1999). "Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I". Mathematical Physics, Analysis and Geometry. 2 (4): 323–415. doi:10.1023/A:1009841100168. ISSN 1385-0172.
8. Ceccherini-Silberstein, Tullio; Krieger, Fabrice; Coornaert, Michel (2014). "An analogue of Fekete's lemma for subadditive functions on cancellative amenable semigroups". J. Anal. Math. 124: 59–81. arXiv:1209.6179. doi:10.1007/s11854-014-0027-4. Theorem 1.1
9. Hille 1948, Theorem 6.6.1. (Measurability is stipulated in Sect. 6.2 "Preliminaries".)
10. Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press. ISBN 978-0-12-622760-4., p.314,12.25
11. Rau-Bredow, H. (2019). "Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures". Risks. 7 (3): 91. doi:10.3390/risks7030091.
12. Shur, Arseny (2012). "Growth properties of power-free languages". Computer Science Review. 6 (5–6): 187–208. doi:10.1016/j.cosrev.2012.09.001.

References

• György Pólya and Gábor Szegő. "Problems and theorems in analysis, volume 1". Springer-Verlag, New York (1976). ISBN 0-387-05672-6.
• Einar Hille. "Functional analysis and semi-groups". American Mathematical Society, New York (1948).
• N.H. Bingham, A.J. Ostaszewski. "Generic subadditive functions." Proceedings of American Mathematical Society, vol. 136, no. 12 (2008), pp. 4257–4266.

External links

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