An alternative interpretation 185
for all weakly moderate t ; i v if X is L L
n
for some weakly moderate n, then 8
X
t + h = 8
X
t +
n
X
k=1
h
k
k Mi X
k
ex pi t X + h
n
ε where ε
✁
0 for all h
✁
0 and weakly moderate t ; v
if S = X
1
, ..., X
n
is a sample of independent random numbers then 8
X
1
+···+ X
n
t
✁
8
X
1
t · · · 8
X
n
t . For the proof we refer to [6].
5. Theoretic model inside the RFS context
To formalize the empirical statistical description of §1 we introduce a random number X
i, j,k
for every cell i, j, k where 1
i n, 1
j p, 1
k q . These random numbers take their
values in a discrete finite set E and the numbers n, p, q are large while s is a very large. The cells are supposed to be of small size. Thus the model can be visualized by a multidimensional
table x
a i, j,k
, 1 a
s. We have then the following cases:
a linear material is represented by a nxs matrix; a bidimensional material is represented by a cubic nx pxs matrix;
a tridimensional material is represented by an hypercubic nx pxqxs matrix. The statistical matter behavior law can be expressed by means of the following conditional
frequencies: f r X
i, j,k
= a | X
i
1
, j
1
, k
1
= a
1
, X
i
2
, j
2
, k
2
= a
2
, · · · , X
i
r
, j
r
, k
r
= a
r
. The model suggests the following rough classification of behaviors:
A Local behaviors: among them we distinguish between the independent case ∀
a ∈ E , ∀a
1
, · · · , a
r
∈ E
f r X
i, j,k
= a | X
i
1
, j
1
, k
1
= a
1
, X
i
2
, j
2
, k
2
= a
2
, · · · , X
i
r
, j
r
, k
r
= a
r
− f r X
i, j,k
= a
≈ and the weakly dependence case expressed by the conditions
∀ a ∈ E , ∀a
1
, · · · , a
r
∈ E
f r X
i, j,k
= a | X
i
1
, j
1
, k
1
= a
1
, X
i
2
, j
2
, k
2
= a
2
, · · · , X
i
r
, j
r
, k
r
= a
r
− f r X
i, j,k
= a
∼ B Non local behavior, where we distinguish between the short range dependence expressed
by the two conditions ∀
a ∈ E , ∀a
1
, · · · , a
r
∈ E
f r X
i, j,k
= a
− f r X
i, j,k
= a | X
i
r
, j
r
, k
r
= a
r
not ≈ 0 f r X
i, j,k
= a | X
i
1
, j
1
, k
1
= a
1
, X
i
2
, j
2
, k
2
= a
2
, · · · , X
i
r
, j
r
, k
r
= a
r
− f r X
i, j,k
= a | X
i
r
, j
r
, k
r
= a
r
≈ and the weak short range dependence case expressed by:
f r X
i, j,k
= a
− f r X
i, j,k
= a | X
i
r
, j
r
, k
r
= a
r
not ≈ 0
186 M. Magno - M. Musio
f r X
i, j,k
= a | X
i
1
, j
1
, k
1
= a
1
, X
i
2
, j
2
, k
2
= a
2
, · · · , X
i
r
, j
r
, k
r
= a
r
− f r X
i, j,k
= a | X
i
r
, j
r
, k
r
= a
r
∼ the cells i
r
, j
r
, k
r
range into the neighborhood of the i, j, kcell; C the long range dependence case expressed by the conditions:
∀ a ∈ E , ∀a
1
, · · · , a
r
∈ E
f r X
i, j,k
= a
− f r X
i, j,k
= a | X
i
r
, j
r
, k
r
= a
r
not ≈ 0 and
f r X
i, j,k
= a
− f r X
i, j,k
= a | X
i
r
, j
r
, k
r
= a
r
where the cells i
r
, j
r
, k
r
are not necessary in the neighborhood of the i, j, kcell. This classification may be refined if one relates the dependences with the distances. The
inversion formula of the characteristic function may be useful to treat the information in order to eliminate the white noise and to put in evidence the intrinsic characteristic distances of the
concerned matter.
6. Conclusion
