Page 103 - Special Topic Session (STS) - Volume 3

STS517 Jan Rosinski´
where

We have

where is the family of all partitions P = {P , . . . , P } of {1, . . . , m}, and |P| is
m
k
1
q .
the number of sets in partition P; q P j = ∏ i∈P j i
In particular, if the supports of q ’s are pairwise disjoint modulo ν, then
i

If the supports of q ’s are triple-wise disjoint modulo ν, then
i

where m,2 is the family of all partitions P = {P , . . . , P } of {1, . . . , m}, such that
1
k
|P| = 1 or 2, |P| is the number of sets in P, and q = ∏ i∈P j q . κ(P) = Card{j ∶
P j
i
j
|P| = 2}.
j

4. Discussion and Conclusion
We illustrate this Theorem 5 on a familiar case of Lévy processes.
Example 7. Let X = (X ) ≥ 0 be a Lévy process such that e iuX t = e tK(u) ,
t t
where K is a cumulant function given by

Let q ∶ ℝ × ℝ ↦ ℝ + be a measurable function such that
+
∫ q(r, v)drρ(dv) = 1.Then for any measurable functional F ∶ ℝ (0,∞) ↦ ℝ
ℝ + ×ℝ

̃
̃
In this example ((, )) = {≤} , where (, ) ∈ Ω = ℝ × ℝ and Ω is
+

equipped with probability measure ℙ(, ) = (, )(). The Lévy
̃
measure of is the “distribution” of the process on ℝ ℝ+ under possibly
infinite infinite measure () on Ω . Therefore, one has as many random
̃
translations of a Lévy process as the number of choices of a nonnegative
function satisfying ∫ (, )() = 1. This is already a rich class.
ℝ + ×ℝ
The possibility of selecting of many independent translations increases their
applicability but that also increases the complexity of () in the change of

92 | I S I W S C 2 0 1 9