Subadditivity

 ( Mathematical Analysis ) 

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. are special cases of subadditive functions.

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Definitions A subadditive function is a function $f\; \backslash colon\; A\; \backslash to\; B$, having a domain A and an partial order codomain B that are both closed under addition, with the following property: $$\backslash forall\; x,\; y\; \backslash in\; A,\; f(x+y)\backslash leq\; f(x)+f(y).$$

An example is the square root function, having the non-negative as domain and codomain: since $\backslash forall\; x,\; y\; \backslash geq\; 0$ we have: $$\backslash sqrt\{x+y\}\backslash leq\; \backslash sqrt\{x\}+\backslash sqrt\{y\}.$$

A sequence $\backslash left\; \backslash \{\; a\_n\; \backslash right\; \backslash \}\_\{n\; \backslash geq\; 1\}$ is called subadditive if it satisfies the inequality $$a\_\{n+m\}\backslash leq\; a\_n+a\_m$$ 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.

Note that while a concave sequence is subadditive, the converse is false. For example, arbitrarily assign $a\_1,\; a\_2,\; ...$ with values in $0.5,$; then the sequence is subadditive but not concave.

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Properties

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Sequences A useful result pertaining to subadditive sequences is the following lemma due to Michael Fekete.

{n_k} > s^* + \epsilon for all $k$.

Since $s^*\; :=\; \backslash inf\_n\; \backslash frac\{a\_n\}n$, there exists an $a\_m$ such that .

By infinitary pigeonhole principle, there exists a sub-subsequence of $(a\_\{n\_k\})\_k$, whose indices all belong to the same residue class modulo $m$, and so they advance by multiples of $m$. This sequence, continued for long enough, would be forced by subadditivity to dip below the $s^*\; +\; \backslash epsilon$ slope line, a contradiction.

In more detail, by subadditivity, we have

which implies , a contradiction.

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

There are extensions of Fekete's lemma that do not require the inequality $a\_\{n+m\}\backslash le\; a\_n\; +\; a\_m$ to hold for all m and n, but only for m and n such that $\backslash frac\; 1\; 2\; \backslash le\; \backslash frac\; m\; n\; \backslash le\; 2.$

Moreover, the condition $a\_\{n+m\}\backslash le\; a\_n\; +\; a\_m$ may be weakened as follows: $a\_\{n+m\}\backslash le\; a\_n\; +\; a\_m\; +\; \backslash phi(n+m)$ provided that $\backslash phi$ is an increasing function such that the integral $\backslash int\; \backslash phi(t)\; t^\{-2\}\; \backslash ,\; dt$ converges (near the infinity). (The same as Indagationes Math. 14.) See also Steele 1997, Theorem 1.9.2.

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 superadditive and subadditivity is present.Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997). .

Additionally, analogues of Fekete's lemma have been proven for subadditive real maps (with additional assumptions) from finite subsets of an amenable group, Theorem 6.1 and further, of a cancellative left-amenable semigroup. Theorem 1.1

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Functions 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. $f(x)\; \backslash ge\; f(x+y)\; -\; f(y)$. Hence $f(0)\; \backslash ge\; f(0+y)\; -\; f(y)\; =\; 0$

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

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The negative of a subadditive function is superadditivity.

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Examples in various domains

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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.

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Economics Subadditivity is an essential property of some particular Cost curve. 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.

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Finance Subadditivity is one of the desirable properties of coherent risk measures in risk management. 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. 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 $V$ at the confidence level $1-p$ is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss, $$\backslash text\{VaR\}\_p\; \backslash equiv\; z\_\{p\}\backslash sigma\_\{\backslash Delta\; V\}\; =\; z\_\{p\}\backslash sqrt\{\backslash sigma\_x^2+\backslash sigma\_y^2+2\backslash rho\_\{xy\}\backslash sigma\_x\; \backslash sigma\_y\}$$ where $z\_p$ is the inverse of the normal cumulative distribution function at probability level $p$, $\backslash sigma\_x^2,\backslash sigma\_y^2$ are the individual positions returns variances and $\backslash rho\_\{xy\}$ is the linear correlation measure between the two individual positions returns. Since variance is always positive, $$\backslash sqrt\{\backslash sigma\_x^2+\backslash sigma\_y^2+2\backslash rho\_\{xy\}\backslash sigma\_x\; \backslash sigma\_y\}\; \backslash leq\; \backslash sigma\_x\; +\; \backslash sigma\_y$$ Thus the Gaussian VaR is subadditive for any value of $\backslash rho\_\{xy\}\; \backslash in\; -1,1$ and, in particular, it equals the sum of the individual risk exposures when $\backslash rho\_\{xy\}=1$ which is the case of no diversification effects on portfolio risk.

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

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

For every $k\; \backslash geq\; 1$, sample two strings of length $n$ uniformly at random on the alphabet $1,\; 2,\; ...,\; k$. The expected length of the longest common subsequence is a super-additive function of $n$, and thus there exists a number $\backslash gamma\_k\; \backslash geq\; 0$, such that the expected length grows as $\backslash sim\; \backslash gamma\_k\; n$. By checking the case with $n=1$, we have . The exact value of even $\backslash gamma\_2$, however, is only known to be between 0.788 and 0.827.

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See also

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Notes

• György Pólya and Gábor Szegő. Problems and Theorems in Analysis, vol. 1. Springer-Verlag, New York (1976). .
• 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.

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External links

Categories: Mathematical Analysis, Sequences And Series, Types Of Functions

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