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waiting-time distribution is hyperbolic, the transition probability density is hyperbolic
or both.
Consider the inverse power-law waiting-time distribution function with no finite
moments and the asymptotic form
1
t
β
+
1
ψ (
t
)
∼
(5.149)
with 0
1
case, it is not possible to simply Taylor expand the Laplace transform of the waiting-
time distribution function. To circumvent this formal difficulty the Laplace transform of
the waiting-time distribution can be written
<β<
1. When the lowest moment of a distribution diverges, as in the
β<
∞
−
1
)
=
1
e
−
ut
ψ(
ψ(
−
ψ(
u
)
=
1
u
1
−
−
t
)
dt
.
(5.150)
0
On introducing the scaling variable
y
=
ut
we obtain for the integral
∞
∞
y
u
dy
u
=
u
β
∞
0
1
e
−
ut
ψ(
1
e
−
y
ψ
1
e
−
y
ψ(
−
t
)
dt
=
−
−
y
)
dy
,
0
0
(5.151)
where the right-most integral, obtained using (
5.149
), is finite for the specified interval
of the scaling index. Therefore we obtain the expansion for the Laplace transform of the
waiting-time distribution function
ψ(
Au
β
,
u
)
=
1
−
(5.152)
so the scaling index is positive definite and
ψ(
0. The Laplace trans-
form of the memory function associated with this Laplace transform waiting-time
distribution, to lowest order in the Laplace variable, is given by
u
)
→
1as
u
→
0
φ(
u
1
−
β
.
lim
u
u
)
∝
(5.153)
→
Consequently, using a Tauberian theorem,
t
n
−
1
(
1
u
n
⇔
)
,
(5.154)
n
which it is simple to show that asymptotically in time the memory function is given by
t
β
−
2
lim
→∞
φ(
t
)
∝
,
(5.155)
t
which is an inverse power law with an index 1
.
This argument can be repeated for the jump probability when it too is no longer
an analytic function, which is to say that the central moments diverge as in the case
above. We consider the situation of a symmetric transition probability
p
<
2
−
β<
2
,
but for simplicity of presentation we restrict the argument to one spatial dimension. In
the context of random walks this probability determines the length of the successive
steps within the walk and here we take it to be given by the inverse power law in the
step length,
(
q
)
=
p
(
−
q
)