Existence of limits of analytic one-parameter semigroups of copulas

doi:10.1285/i15900932v30n2p1

Existence of limits of analytic one-parameter semigroups of copulas

William F. Darsow, Elwood T. Olsen

Abstract

A 2-copula $F$ is idempotent if $F*F=F$ . Here $*$ denotes the product defined in [1]. An idempotent copula $F$ is said to be a unit for a 2-copula $A$ if $F*A=A*F=A$ . An idempotent copula is said to annihilate a 2-copula $A$ if $F*A=A*F=F$ .
If $F$ is a unit for $A$ and $s$ is a non-negative real number, define

$\text{[math]}$

For any copula $A$ and any idempotent copula $F$ which is a unit for $A$ , the set

$\text{[math]}$

is a semigroup of copulas under the $*$ operation, which is homomorphic to the semigroup $[0,\infty )$ under addition. We call this set an analyticone-parameter semigroup of copulas. $C_s$ can be defined also for $s<0$ , and $C_{-s}*C_s=C_s*C_{-s}=F$ , but in general $C_s$ is not a copula for $s<0$ .
We show that for any such analytic one-parameter semigroup, the limit $\lim_{s\to \infty}C_s=E$ exists. We show also that the limit $E$ has the followingproperties:
(i) $E$ is idempotent.
(ii) $E$ annihilates $A$ , $F$ and $C_s$ .
(iii) $E$ is the greatest annihilator of $A$ and of $C_s$ , $s\in (0,\infty )$ .
\noindent It is also true that $F$ is the least unit for $C_s$ , $s\in [0,\infty)$ . We give a geometrical interpretation of this result, and we comment on theuse of analytic semigroups to construct Markov processes with continuousparameter.

DOI Code: 10.1285/i15900932v30n2p1

Keywords: copula; idempotent; star product

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Existence of limits of analytic one-parameter semigroups of copulas

Abstract