2.2 Properties of the corresponding process
From the previous results we can conclude that the GML function E
k+1 ν,ν k+1
−λt
ν
, k ≥ 0, suitably normalized by the factor
λt
ν k
, represents a proper probability distribution and we can indicate it as Pr
N
ν
t = k .
Moreover by 2.10 we can consider the process N
ν
t, t 0 as a time-changed Poisson process. It is well-known see [12] that, for a homogeneous Poisson process N subject to a random time
change by the random function Λ0, t], the following equality in distribution holds:
N Λ0, t]
d
= M t, 2.15
where M t, t 0 is a Cox process directed by Λ. In our case the random measure Λ0, t] possesses
distribution v
2 ν
given in 2.9 with 2.11 and we can conclude that N
ν
is a Cox process. This conclusion will be confirmed by the analysis of the factorial moments.
Moreover, as remarked in [2] and [18], the fractional Poisson process N
ν
t, t 0 represents a renewal process with interarrival times U
j
distributed according to the following density, for j = 1, 2, ...:
f
ν 1
t = Pr ¦
U
j
∈ d t ©
d t = λt
ν−1
E
ν,ν
−λt
ν
, 2.16
with Laplace transform L
¦ f
ν 1
t; s ©
= λ
s
ν
+ λ .
2.17 Therefore the density of the waiting time of the k-th event, T
k
= P
k j=1
U
j
, possesses the Laplace transform
L ¦
f
ν k
t; s ©
= λ
k
s
ν
+ λ
k
. 2.18
Its inverse can be obtained by applying again 2.7 for β = ν, γ = ν k and ω = −λ and can be
expressed, as for the probability distribution, in terms of a GML function as f
ν k
t = Pr T
k
∈ d t d t = λ
k
t
ν k−1
E
k ν,ν k
−λt
ν
. 2.19
The corresponding distribution function can be obtained by integrating 2.19 F
ν k
t = Pr T
k
t 2.20
= λ
k
Z
t
s
ν k−1 ∞
X
j=0
k − 1 + j−λs
ν j
jk − 1Γν j + ν k
ds =
λ
k
t
ν k
ν
∞
X
j=0
k − 1 + j−λt
ν j
jk − 1k + jΓν j + ν k
= λ
k
t
ν k ∞
X
j=0
k − 1 + j−λt
ν j
jk − 1Γν j + ν k + 1
= λ
k
t
ν k
E
k ν,ν k+1
−λt
ν
. We can check that 2.20 satisfies the following relationship
Pr T
k
t − Pr
T
k+1
t = p
ν k
t, 2.21
690
for p
ν k
given in 2.5. Indeed from 2.20 we can rewrite 2.21 as λ
k
t
ν k
E
k ν,ν k+1
−λt
ν
− λ
k+1
t
νk+1
E
k+1 ν,νk+1+1
−λt
ν
= λ
k
t
ν k ∞
X
j=0
k − 1 + j−λt
ν j
jk − 1Γν j + ν k + 1
− λ
k+1
t
νk+1 ∞
X
j=0
k + j−λt
ν j
jkΓ ν j + ν k + ν + 1
= by putting l = j + 1 in the second sum
= λ
k
t
ν k ∞
X
j=0
k − 1 + j−λt
ν j
jk − 1Γν j + ν k + 1
+ λ
k
t
ν k ∞
X
l=1
k + l − 1−λt
ν l
l − 1kΓν l + ν k + 1 =
λ
k
t
ν k ∞
X
j=0
k + j−λt
ν j
jkΓ ν j + ν k + 1
= p
ν k
t.
Remark 2.3 As pointed out in [18] and [25], the density of the interarrival times 2.16 possess the following asymptotic behavior, for t
→ ∞: Pr
¦ U
j
∈ d t ©
d t =
λt
ν−1
E
ν,ν
−λt
ν
= − d
d t E
ν,1
−λt
ν
2.22 =
λ
1 ν
sin νπ
π Z
+∞
r
ν
e
−λ
1 ν
r t
r
2 ν
+ 2r
ν
cos νπ + 1
d r ∼
sin νπ
π Γν + 1
λt
ν+1
= ν
λΓ1 − νt
ν+1
, where the well-known expansion of the Mittag-Leffler function given in 5.3 has been applied. The
density 2.22 is characterized by fat tails with polynomial, instead of exponential, decay and, as a consequence, the mean waiting time is infinite.
For t → 0 the density of the interarrival times displays the following behavior:
Pr ¦
U
j
∈ d t ©
d t ∼ λt
ν−1
Γν ,
2.23 which means that
U
j
takes small values with large probability. Therefore, by considering 2.22 and 2.23 together, we can draw the conclusion that the behavior of the density of the interarrival times
differs from standard Poisson in that the intermediate values are assumed with smaller probability than in the exponential case.
Remark 2.4 We observe that also for the waiting-time density 2.19 we can find a link with the solution to the fractional diffusion equation 2.8. This can be shown by rewriting its Laplace transform 2.18
as
L ¦
f
ν k
t; s ©
= λ
k
s
ν
+ λ
k
= Z
+∞
e
−s
ν
t
λ
k
t
k −1
k − 1 e
−λt
d t. By recalling that
e
−s
ν
y λ
= Z
+∞
e
−sz
g
ν
z; y
λ dz,
ν 1, y 0, 2.24
691
for the stable law g
ν
·; y defined above, we get f
ν k
t = Z
+∞
g
ν
t; y
λ y
k −1
e
− y
k − 1 d y.
2.25 Formula 2.25 permits us to conclude that f
ν k
t can be interpreted as the law of the stable random variable S
ν
with a random scale parameter possessing an Erlang distribution.
2.3 The probability generating function
Kategori