probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

and

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, subindependence is a weak form of

independence Independence is a condition of a person, nation, country, or state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the statu ...

. Two

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

s ''X'' and ''Y'' are said to be subindependent if the

characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...

of their sum is equal to the product of their marginal characteristic functions. Symbolically: :

\varphi_(t) = \varphi_X(t)\cdot\varphi_Y(t). \,

This is a weakening of the concept of independence of random variables, i.e. if two random variables are independent then they are subindependent, but not conversely. If two random variables are subindependent, and if their covariance exists, then they are

uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, ther ...

.Hamedani & Volkmer (2009) Subindependence has some peculiar properties: for example, there exist random variables ''X'' and ''Y'' that are subindependent, but ''X'' and ''αY'' are not subindependent when ''α'' ≠ 1 and therefore ''X'' and ''Y'' are not independent. One instance of subindependence is when a random variable ''X'' is

Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...

with location 0 and scale ''s'' and another random variable ''Y''=''X'', the antithesis of independence. Then ''X+Y'' is also Cauchy but with scale ''2s''. The characteristic function of either ''X'' or ''Y'' in ''t'' is then ''exp''(-''s''·, ''t'', ), and the characteristic function of ''X+Y'' is ''exp''(-2''s''·, ''t'', )=''exp''(-''s''·, ''t'', )².

Notes

References

Further reading