Existence of limits of analytic one-parameter semigroups of copulas
Abstract
A 2-copula
is idempotent if
. Here
denotes the product defined in [1]. An idempotent copula
is said to be a unit for a 2-copula
if
. An idempotent copula is said to annihilate a 2-copula
if
.
If
is a unit for
and
is a non-negative real number, define
and any idempotent copula
which is a unit for
, the set
operation, which is homomorphic to the semigroup
under addition. We call this set an analyticone-parameter semigroup of copulas.
can be defined also for
, and
, but in general
is not a copula for
.
We show that for any such analytic one-parameter semigroup, the limit
exists. We show also that the limit
has the followingproperties:
(i)
is idempotent.
(ii)
annihilates
,
and
.
(iii)
is the greatest annihilator of
and of
,
.
\noindent It is also true that
is the least unit for
,
. We give a geometrical interpretation of this result, and we comment on theuse of analytic semigroups to construct Markov processes with continuousparameter.
![F](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
![F*F=F](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/08626eb37ca8c2036f36db573bf460d7.png)
![*](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/3389dae361af79b04c9c8e7057f60cc6.png)
![F](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
![A](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/7fc56270e7a70fa81a5935b72eacbe29.png)
![F*A=A*F=A](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/6226f98b0b324cf738a98fe0880f16fe.png)
![A](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/7fc56270e7a70fa81a5935b72eacbe29.png)
![F*A=A*F=F](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/3524fc57367932c2564341311b8dfda3.png)
If
![F](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
![A](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/7fc56270e7a70fa81a5935b72eacbe29.png)
![s](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/03c7c0ace395d80182db07ae2c30f034.png)
For any copula
![A](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/7fc56270e7a70fa81a5935b72eacbe29.png)
![F](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
![A](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/7fc56270e7a70fa81a5935b72eacbe29.png)
is a semigroup of copulas under the
![*](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/3389dae361af79b04c9c8e7057f60cc6.png)
![[0,\infty )](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/b14ef532a91b6a826c93e81adefa66cd.png)
![C_s](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/2d838fc2872825c85aa6700590019ee2.png)
![s<0](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/67c2c82b33b777833c6d6c718405b350.png)
![C_{-s}*C_s=C_s*C_{-s}=F](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/83e64cb25465882349d4e24cf7bb60c6.png)
![C_s](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/2d838fc2872825c85aa6700590019ee2.png)
![s<0](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/67c2c82b33b777833c6d6c718405b350.png)
We show that for any such analytic one-parameter semigroup, the limit
![\lim_{s\to \infty}C_s=E](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/851e565b16bbdc5793e1a0c2c5e52482.png)
![E](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/3a3ea00cfc35332cedf6e5e9a32e94da.png)
(i)
![E](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/3a3ea00cfc35332cedf6e5e9a32e94da.png)
(ii)
![E](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/3a3ea00cfc35332cedf6e5e9a32e94da.png)
![A](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/7fc56270e7a70fa81a5935b72eacbe29.png)
![F](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
![C_s](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/2d838fc2872825c85aa6700590019ee2.png)
(iii)
![E](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/3a3ea00cfc35332cedf6e5e9a32e94da.png)
![A](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/7fc56270e7a70fa81a5935b72eacbe29.png)
![C_s](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/2d838fc2872825c85aa6700590019ee2.png)
![s\in (0,\infty )](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/1bba22396f4c2e0ff180ff9501a99750.png)
\noindent It is also true that
![F](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/800618943025315f869e4e1f09471012.png)
![C_s](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/2d838fc2872825c85aa6700590019ee2.png)
![s\in [0,\infty)](http://siba-ese.unisalento.it/plugins/generic/latexRender/cache/60151ffd30bd9864c3003b6a4b1c3da9.png)
DOI Code:
10.1285/i15900932v30n2p1
Keywords:
copula; idempotent; star product
Full Text: PDF