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B
| q | α + 1 ,
p
(
q
) =
(5.156)
just as obtained earlier for the Weierstrass walk. A random-walk process in which each
of the steps is instantaneous, but with ( 5.156 ) giving the probability of a step of length
q occurring, gives rise to a limit distribution first discussed by Paul Lévy, that being the
Lévy distribution [ 11 ], which we mentioned previously.
As Weiss [ 29 ] points out, knowledge of the asymptotic behavior of the transition
probability in ( 5.156 ) is sufficient for us to calculate a single correction term to the
normalization condition for the characteristic function (
p
(
k
=
0
) =
1) in the Taylor
expansion of the characteristic function about k
=
0. We proceed as with the waiting-
time distribution and write
1
) =
p
(
k
) =
1
p
(
k
1
[
1
cos
(
kq
) ]
p
(
q
)
dq
,
(5.157)
−∞
where we have used the symmetry properties of the probability distribution to replace
the exponential with the cosine. We are most interested in the region around k
0, but
it is clear that we cannot expand the cosine since the second moment of the distribution
diverges. We can use the same form of transformation of variables y
=
=
kq as used
earlier to write the integral term as
p y
k
dy
k .
1
) p
(
(
)
=
[
]
cos
kq
q
dq
1
cos y
(5.158)
−∞
−∞
Here again the asymptotic form of the distribution dominates the small- k behavior,
so we can write
| α
−∞
1
) p
1
cos y
cos
(
kq
(
q
)
dq
=
B
|
k
dy
.
(5.159)
| α + 1
|
y
−∞
The integral on the rhs of ( 5.159 ) converges for 0
1, so that the leading term in
the expansion of the characteristic function for the steps in the random walk given by
the inverse power law yields
<α<
| α ,
p
(
k
)
1
C
|
k
(5.160)
where the constant C is the product of B and the finite integral to the right in ( 5.159 ).
The demonstration given here is for an inverse power-law index in the interval 0
<α<
1, but the first-order correction to the characteristic function ( 5.160 ) can be shown to be
valid for the interval 0
2; see, for example, the discussion in Weiss [ 29 ].
Note that we now have first-order expansions for the Laplace transform of the
waiting-time distribution function ( 5.152 ) and the Fourier transform of the jump proba-
bility ( 5.160 ) with non-integer exponents in the expansion variables. In general these are
approximate expressions because we can no longer use the Taylor expansion due to the
divergence of the central moments of the corresponding distributions. The correspond-
ing probability density function is then given by the inverse Fourier-Laplace transform
of the equation resulting from inserting ( 5.152 ) and ( 5.160 )into( 5.135 ) to obtain
<α<
u β 1
u β +
P (
k
,
u
)
| α ,
(5.161)
C |
k
 
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