View source for Subadditivity

{{Short description|Property of some mathematical functions}}
In [[mathematics]], '''subadditivity''' is a property of a function that states, roughly, that evaluating the function for the sum of two [[element (set)|elements]] of the [[Domain of a function|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 [[norm (mathematics)|norms]] and [[square roots]]. [[Additive map]]s are special cases of subadditive functions.

==Definitions==
A subadditive function is a [[function (mathematics)|function]] <math>f \colon A \to B</math>, having a [[Domain of a function|domain]] ''A'' and an [[partial order|ordered]] [[codomain]] ''B'' that are both [[closure (mathematics)|closed]] under addition, with the following property:
<math display="block">\forall x, y \in A, f(x+y)\leq f(x)+f(y).</math>

An example is the [[square root]] function, having the [[non-negative]] [[real number]]s as domain and codomain:
since <math>\forall x, y \geq 0</math> we have:
<math display="block">\sqrt{x+y}\leq \sqrt{x}+\sqrt{y}.</math>

A [[sequence]] <math>\left \{ a_n \right \}_{n \geq 1}</math> is called '''subadditive''' if it satisfies the [[inequality (mathematics)|inequality]]
<math display="block"> a_{n+m}\leq a_n+a_m</math>
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, randomly assign <math>a_1, a_2, ...</math> with values in <math>[0.5, 1]</math>; then the sequence is subadditive but not concave.

==Properties==

===Sequences===
{{Anchor|Fekete's Subadditive Lemma}}A useful result pertaining to subadditive sequences is the following [[lemma (mathematics)|lemma]] due to [[Michael Fekete]].<ref>{{cite journal |last=Fekete |first=M. |title=Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten |journal=Mathematische Zeitschrift |volume=17 |issue=1 |year=1923 |pages=228–249 |doi=10.1007/BF01504345 |s2cid=186223729 }}</ref>

{{math theorem|name=Fekete's Subadditive Lemma|math_statement= For every subadditive sequence <math>{\left \{ a_n \right \}}_{n=1}^\infty</math>, the [[Limit of a sequence|limit]] <math>\displaystyle \lim_{n \to \infty} \frac{a_n}{n}</math> is equal to the [[Infimum and supremum|infimum]] <math>\inf \frac{a_n}{n}</math>.  (The limit may be <math>-\infty</math>.)}}

{{Math proof|proof= 
Let <math>s^* := \inf_n \frac{a_n}n</math>.

By definition, <math>\liminf_n \frac{a_n}n \geq s^*</math>. So it suffices to show <math>\limsup_n \frac{a_n}n \leq s^*</math>.

If not, then there exists a subsequence <math>(a_{n_k})_k</math>, and an <math>\epsilon > 0</math>, such that <math>\frac{a_{n_k}}{n_k} > s^* + \epsilon</math> for all <math>k</math>. 

Since <math>s^* := \inf_n \frac{a_n}n</math>, there exists an <math>a_m</math> such that <math>\frac{a_m}m < s^* + \epsilon</math>.

By [[pigeonhole principle#infinite sets|infinitary pigeonhole principle]], there exists a sub-subsequence <math>(a_{n_k})_k</math>, whose indices all belong to the same [[modular arithmetic|residue class]] modulo <math>m</math>, and so they advance by multiples of <math>m</math>. This sequence, continued for long enough, would be forced by subadditivity to dip below the <math>s^* + \epsilon</math> slope line, a contradiction.  

In more detail, by subadditivity, we have

<math display="inline">\begin{aligned} a_{n_2} &\leq a_{n_1} + a_m (n_2-n_1)/m \\ a_{n_3} &\leq a_{n_2} + a_m (n_3-n_2)/m \leq a_{n_1} + a_m (n_3-n_1)/m \\ \cdots & \cdots\\ a_{n_k} &\leq a_{n_1} + a_m (n_k-n_1)/m \end{aligned}</math>

which implies <math>\limsup_k a_{n_k}/n_k \leq a_m/m < s^* + \epsilon</math>, a contradiction.
}}

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

There are extensions of Fekete's lemma that do not require the inequality <math>a_{n+m}\le a_n + a_m</math> to hold for all ''m'' and ''n'', but only for ''m'' and ''n'' such that <math display="inline">\frac 1 2 \le \frac m n \le 2.</math> 

{{Math proof|proof= 

Continue the proof as before, until we have just used the infinite pigeonhole principle.  

Consider the sequence <math>a_m, a_{2m}, a_{3m}, ...</math>. Since <math>2m/m = 2</math>, we have <math>a_{2m} \leq 2a_m</math>. Similarly, we have <math>a_{3m} \leq a_{2m}+a_m \leq 3a_m</math>, etc.  

By the assumption, for any <math>s, t \in \N</math>, we can use subadditivity on them if   

<math display="block">\ln(s+t) \in [\ln(1.5 s), \ln (3s)] = \ln s + [\ln 1.5, \ln 3]</math>

If we were dealing with continuous variables, then we can use subadditivity to go from <math>a_{n_k}</math> to <math>a_{n_k} + [\ln 1.5, \ln 3]</math>, then to <math>a_{n_k} + \ln 1.5 + [\ln 1.5, \ln 3]</math>, and so on, which covers the entire interval <math>a_{n_k} + [\ln 1.5, +\infty)</math>.  

Though we don't have continuous variables, we can still cover enough integers to complete the proof. Let <math>n_k</math> be large enough, such that   

<math display="block">\ln (2) > \ln(1.5) + \ln \left(\frac{1.5 n_k + m}{1.5 n_k}\right) </math>
then let <math>n'</math> be the smallest number in the intersection <math>(n_k + m\Z) \cap (\ln n_k + [\ln(1.5), \ln (3)])</math>. By the assumption on <math>n_k</math>, it's easy to see (draw a picture) that the intervals <math>\ln n_k + [\ln(1.5), \ln (3)]</math> and <math>\ln n' + [\ln(1.5), \ln (3)]</math> touch in the middle. Thus, by repeating this process, we cover the entirety of <math>(n_k + m\Z) \cap (\ln n_k + [\ln(1.5), \infty])</math>.  

With that, all <math>a_{n_k}, a_{n_{k+1}}, ...</math> are forced down as in the previous proof. }}

Moreover, the condition <math>a_{n+m}\le a_n + a_m</math> may be weakened as follows: <math>a_{n+m}\le a_n + a_m + \phi(n+m)</math> provided that <math>\phi</math> is an increasing function such that the integral <math display="inline">\int \phi(t) t^{-2} \, dt</math> converges (near the infinity).<ref>{{cite journal |last1=de Bruijn |first1=N.G. |last2=Erdös |first2=P. |title=Some linear and some quadratic recursion formulas. II |journal=Nederl. Akad. Wetensch. Proc. Ser. A |volume=55 |year=1952 |pages=152–163|doi=10.1016/S1385-7258(52)50021-0 }} (The same as ''Indagationes Math.'' '''14'''.) See also Steele 1997, Theorem 1.9.2.</ref>

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|superadditivity]] and subadditivity is present.<ref>Michael J. Steele. "Probability theory and combinatorial optimization". SIAM, Philadelphia (1997). {{isbn|0-89871-380-3}}.</ref><ref>{{cite video|author=Michael J. Steele|title=CBMS Lectures on Probability Theory and Combinatorial Optimization|publisher=University of Cambridge|year=2011|url=http://sms.cam.ac.uk/collection/1189351}}</ref>

Besides, analogues of Fekete's lemma have been proved for subadditive real maps (with additional assumptions) from finite subsets of an amenable group <ref>{{Cite journal| doi = 10.1007/BF02810577 | doi-access=free | issn = 0021-2172| volume = 115| issue = 1| pages = 1–24| last1 = Lindenstrauss| first1 = Elon| authorlink1=Elon Lindenstrauss | last2 = Weiss| first2 = Benjamin| authorlink2=Benjamin Weiss| title = Mean topological dimension| journal = [[Israel Journal of Mathematics]]| year = 2000| citeseerx = 10.1.1.30.3552}} Theorem 6.1</ref>
<ref>{{Cite journal| doi = 10.1007/BF02790325| doi-access=free | issn = 0021-7670| volume = 48| issue = 1| pages = 1–141| last1 = Ornstein| first1 = Donald S.|authorlink1=Donald Samuel Ornstein| last2 = Weiss| first2 = Benjamin| authorlink2=Benjamin Weiss| title = Entropy and isomorphism theorems for actions of amenable groups| journal = [[Journal d'Analyse Mathématique]]| year = 1987}}</ref>
,<ref>{{Cite journal| doi = 10.1023/A:1009841100168| issn = 1385-0172| volume = 2| issue = 4| pages = 323–415| last = Gromov| first = Misha| title = Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I| journal = Mathematical Physics, Analysis and Geometry| year = 1999| bibcode = 1999MPAG....2..323G| s2cid = 117100302}}</ref>
and further, of a cancellative left-amenable semigroup.<ref>{{Cite journal | arxiv = 1209.6179| last1 = Ceccherini-Silberstein| first1 = Tullio| title = An analogue of Fekete's lemma for subadditive functions on cancellative amenable semigroups| journal = [[Journal d'Analyse Mathématique]]| volume = 124| pages = 59–81| last2 = Krieger| first2 = Fabrice| last3 = Coornaert| first3 = Michel| year = 2014| doi = 10.1007/s11854-014-0027-4 | doi-access=free}} Theorem 1.1</ref>

===Functions===
{{math theorem|name='''Theorem:'''<ref>Hille 1948, Theorem 6.6.1. (Measurability is stipulated in Sect. 6.2 "Preliminaries".)</ref>|math_statement= For every [[Measurable function|measurable]] subadditive function <math>f : (0,\infty) \to \R,</math> the limit <math>\lim_{t\to\infty} \frac{f(t)}{t}</math> exists and is equal to <math>\inf_{t>0} \frac{f(t)}{t}.</math> (The limit may be <math>-\infty.</math>)}}

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. <math>f(x) \ge f(x+y) - f(y)</math>. Hence <math>f(0) \ge f(0+y) - f(y) = 0</math>

A [[concave function]] <math>f: [0,\infty) \to \mathbb{R}</math> with <math>f(0) \ge 0</math> is also subadditive.
To see this, one first observes that <math>f(x) \ge \textstyle{\frac{y}{x+y}} f(0) + \textstyle{\frac{x}{x+y}} f(x+y)</math>.
Then looking at the sum of this bound for <math>f(x)</math> and <math>f(y)</math>, will finally verify that ''f'' is subadditive.<ref>{{cite book | last = Schechter | first=Eric | author-link=Eric Schechter| title=Handbook of Analysis and its Foundations | publisher=Academic Press | location = San Diego | year=1997 | isbn=978-0-12-622760-4}}, p.314,12.25</ref>

The negative of a subadditive function is [[superadditivity|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|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 [[Strong subadditivity of quantum entropy|quantum analog]].

===Economics===

Subadditivity is an essential property of some particular [[Cost curve|cost function]]s. 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.{{citation needed|date=October 2015}}

===Finance===
Subadditivity is one of the desirable properties of [[coherent risk measure]]s in [[risk management]].<ref name="Rau-Bredow">{{Cite journal | doi = 10.3390/risks7030091| title = Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures| year = 2019| last1 = Rau-Bredow | first1 = H. | journal = Risks| volume = 7| issue = 3|pages = 91| doi-access = free| hdl = 10419/257929| hdl-access = free}}</ref> 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 [[Value at risk|VaR]] models which do not rely on the assumption of [[Normal distribution|normality]] of risk factors. The Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio <math> V </math> at the confidence level <math> 1-p </math> is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss,
<math display="block"> \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} </math>
where <math> z_p </math> is the inverse of the normal [[cumulative distribution function]] at probability level <math> p </math>, <math> \sigma_x^2,\sigma_y^2 </math> are the individual positions returns variances and <math> \rho_{xy} </math> is the [[Pearson correlation coefficient|linear correlation measure]] between the two individual positions returns. Since [[variance]] is always positive,
<math display="block"> \sqrt{\sigma_x^2+\sigma_y^2+2\rho_{xy}\sigma_x \sigma_y} \leq \sigma_x + \sigma_y  </math>
Thus the Gaussian VaR is subadditive for any value of <math> \rho_{xy} \in [-1,1] </math> and, in particular, it equals the sum of the individual risk exposures when <math> \rho_{xy}=1 </math> which is the case of no diversification effects on portfolio risk.

===Thermodynamics===
Subadditivity occurs in the thermodynamic properties of non-[[ideal solution]]s and mixtures like the excess molar volume and [[heat of mixing]] or excess enthalpy.

===Combinatorics on words===
A factorial [[Formal language|language]] <math>L</math> is one where if a [[String (computer science)|word]] is in <math>L</math>, then all [[Substring|factors]] of that word are also in <math>L</math>.   In combinatorics on words, a common problem is to determine the number <math>A(n)</math> of length-<math>n</math> words in a factorial language.  Clearly <math>A(m+n) \leq A(m)A(n)</math>, so <math>\log A(n)</math> is subadditive, and hence Fekete's lemma can be used to estimate the growth of <math>A(n)</math>.<ref name=shur>{{cite journal|last=Shur|first=Arseny|title=Growth properties of power-free languages|journal=Computer Science Review|date=2012|volume=6|issue=5–6|pages=187–208|doi=10.1016/j.cosrev.2012.09.001}}</ref>

For every <math>k \geq 1</math>, sample two strings of length <math>n</math> uniformly at random on the alphabet <math>1, 2, ..., k</math>. The expected length of the [[longest common subsequence]] is a ''super''-additive function of <math>n</math>, and thus there exists a number <math>\gamma_k \geq 0</math>, such that the expected length grows as <math>\sim \gamma_k n</math>. By checking the case with <math>n=1</math>, we easily have <math>\frac 1k < \gamma_k \leq 1</math>. The exact value of even <math>\gamma_2</math>, however, is only known to be between 0.788 and 0.827.<ref>{{Cite journal |last=Lueker |first=George S. |date=May 2009 |title=Improved bounds on the average length of longest common subsequences |url=https://dl.acm.org/doi/10.1145/1516512.1516519 |journal=Journal of the ACM |language=en |volume=56 |issue=3 |pages=1–38 |doi=10.1145/1516512.1516519 |s2cid=7232681 |issn=0004-5411}}</ref>

== See also ==

* {{annotated link|Apparent molar property}}
* {{annotated link|Choquet integral}}
* {{annotated link|Superadditivity}}
* {{annotated link|Triangle inequality}}

==Notes==
<references />

==References==
*[[György Pólya]] and [[Gábor Szegő]]. ''[[Problems and Theorems in Analysis]]'', vol. 1. Springer-Verlag, New York (1976). {{isbn|0-387-05672-6}}.
*[[Einar Hille]]. "[https://archive.org/details/functionalanalys017173mbp 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.&nbsp;4257–4266.

==External links==
{{PlanetMath attribution|id=4615|title=subadditivity}}

[[Category:Mathematical analysis]]
[[Category:Sequences and series]]
[[Category:Types of functions]]