Existence of limits of analytic one-parameter semigroups of copulas


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
For any copula A and any idempotent copula F which is a unit for A, the set
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, andC_{-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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