DOC.
30
QUASIPERIODIC
PROCESSES
303
characteristic also
represents an expression
of the statistical character
peculiar
to
F;
for
it
expresses
the
degree
of
dependence existing
between
Fvalues
whose
arguments
differ
by
A. It
will
turn out
that there exists
a
simple dependence
between the
"characteristic" and the
"intensity
curve."
It
follows
from the
meaning
of
X
that
it
is
not
necessary
to
choose
exactly
T
as [p. 1]
the
integration
interval
in
(2);
any sufficiently large
tinterval could be chosen
instead.
In
the
case
of the choice made
in
(2),
one
has
to
conceive of
F(t)
as
being
continued
over
t
=
T to
t
=
T
+
A
in
such
a way
that for
this
piece
one
sets
F(T
+
t')
=
F(t').
We do this
in
order
to
be able
to
apply
the
expansion
(1)
without
gaps.
This
expedient
is
without
practical significance
because
X(A)
will
depend
on
A only
for
A
that
are so
small that
A
is
small
compared
with
T.
If this
were
not
the
case,
then the whole observational interval
T
would
not
be
sufficiently large
for
a
statistical
investigation.
If
we
insert
(1)
into
(2), we
obtain
A
2
oo
2vnA
...(3)
2x(A)
=

+
£
A
n
cos
z
1
1
From this
one can
discern
at
once
the close connection between the
intensity
function
and
the
characteristic.
For,
since the cosine in
(3)
varies
slowly
with
n
because of the
smallness of
A/T,
one can
immediately replace
A2n
by
the
mean
value
A2n
character
ized above.
Furthermore,
we
introduce the
quantity 2nn/T
=
X
in
place
of
n;
when
n
runs
through
the
integers,
xA
runs
through
values that differ from
one
another
very
little,
namely by
Adx
=
2xA/T.
We
can,
therefore, convert
the
sum
in
(3)
into the
integral
oo
2%
(A) =
const
+
fAlcos(xA)ds
o
277
or,
if
we
set
7
A"2 =
I(X),
2tt
which
quantity
is
independent
of the
magnitude
of
T
and shall be called the
spectral
"intensity,"
oo
2%(A)
=
const
+
Jt/(x)cos(xA)dx
...(3a)
o
where
x
is
a
frequency multiplied
by
2n and referred
to
the unit of
length
of the
t
axis.