<s>
In	O
mathematics	O
,	O
the	O
convolution	B-Algorithm
power	I-Algorithm
is	O
the	O
n-fold	O
iteration	O
of	O
the	O
convolution	B-Language
with	O
itself	O
.	O
</s>
<s>
where	O
∗	O
denotes	O
the	O
convolution	B-Language
operation	O
of	O
functions	O
on	O
Rd	O
and	O
δ0	O
is	O
the	O
Dirac	O
delta	O
distribution	O
.	O
</s>
<s>
If	O
x	O
is	O
the	O
distribution	O
function	O
of	O
a	O
random	O
variable	O
on	O
the	O
real	O
line	O
,	O
then	O
the	O
nth	O
convolution	B-Algorithm
power	I-Algorithm
of	O
x	O
gives	O
the	O
distribution	O
function	O
of	O
the	O
sum	O
of	O
n	O
independent	O
random	O
variables	O
with	O
identical	O
distribution	O
x	O
.	O
</s>
<s>
Intuitively	O
,	O
a	O
measure	O
should	O
be	O
infinitely	O
divisible	O
provided	O
it	O
has	O
a	O
well-defined	O
"	O
convolution	B-Language
logarithm.	O
"	O
</s>
<s>
Many	O
applications	O
of	O
the	O
convolution	B-Algorithm
power	I-Algorithm
rely	O
on	O
being	O
able	O
to	O
define	O
the	O
analog	O
of	O
analytic	B-Language
functions	I-Language
as	O
formal	O
power	O
series	O
with	O
powers	O
replaced	O
instead	O
by	O
the	O
convolution	B-Algorithm
power	I-Algorithm
.	O
</s>
<s>
where	O
denotes	O
the	O
derivative	B-Algorithm
operator	O
.	O
</s>
<s>
Specifically	O
,	O
this	O
holds	O
if	O
x	O
is	O
a	O
compactly	O
supported	O
distribution	O
or	O
lies	O
in	O
the	O
Sobolev	O
space	O
W1	O
,	O
1	O
to	O
ensure	O
that	O
the	O
derivative	B-Algorithm
is	O
sufficiently	O
regular	O
for	O
the	O
convolution	B-Language
to	O
be	O
well-defined	O
.	O
</s>
<s>
In	O
the	O
configuration	O
random	O
graph	O
,	O
the	O
size	O
distribution	O
of	O
connected	O
components	O
can	O
be	O
expressed	O
via	O
the	O
convolution	B-Algorithm
power	I-Algorithm
of	O
the	O
excess	O
degree	O
distribution	O
(	O
)	O
:	O
</s>
<s>
As	O
convolution	B-Language
algebras	O
are	O
special	O
cases	O
of	O
Hopf	O
algebras	O
,	O
the	O
convolution	B-Algorithm
power	I-Algorithm
is	O
a	O
special	O
case	O
of	O
the	O
(	O
ordinary	O
)	O
power	O
in	O
a	O
Hopf	O
algebra	O
.	O
</s>
<s>
In	O
applications	O
to	O
quantum	O
field	O
theory	O
,	O
the	O
convolution	B-Language
exponential	O
,	O
convolution	B-Language
logarithm	O
,	O
and	O
other	O
analytic	B-Language
functions	I-Language
based	O
on	O
the	O
convolution	B-Language
are	O
constructed	O
as	O
formal	O
power	O
series	O
in	O
the	O
elements	O
of	O
the	O
algebra	O
.	O
</s>
