Page 101 - Special Topic Session (STS) - Volume 3

STS517 Jan Rosinski´
(L1) for every t ∈ T ∫ |x(t)| ⋀1ν(dx) < ∞,
2
T
ℝ
(L2) for every A ∈ v(A) = v ∗ (A\O ), where ν∗ is the inner measure.
T
T
If only condition (L1) is assumed, then ν is called a pre Lévy measure.
Condition (L1) is a technical one, needed for the integral in the Lévy-
Khintchine formula (3) to be well-defined. Condition (L2) gives a rigorous
meaning to “ν has no mass at the origin”. Indeed, if T is countable, then ∈

and (L2) is equivalent to ν(0T) = 0, which is the usual condition for Lévy

measures. If T is uncountable, then ∉ , so that ν(0T) is undefined.

However, (L2) still makes sense and it ensures uniqueness of ν. Indeed, we will
show that every infinitely divisible process has a unique Lévy measure
satisfying the above definition.
We will now give the Lévy-Khintchine representation for an arbitrary
infinitely divisible process obtained in Rosinski, J. (2018). The key to this
representation is the Definition 1,´ which encompassed any infinitely divisible
process. Special cases of the representation, under additional assumptions on
the underlying process, were obtained in Barndorff-Nielsen et al. (2015) and
Kabluchko, Z., &Stoev, S. (2016). Below and in what follows, Tˆ will denote the
family of all finite nonempty subsets of the index set T,

′
̂
so that for any ∈ , ℝ can be identified with the Euclidean space ℝ () . ⟦. ⟧
will denote a fixed truncation functions, see Rosinski, J. (2018) for details.´
Theorem 2. Let X = (X ) be an infinitely divisible process. Then there exist
t t∈T
a unique triplet (∑, v, b) consisting of a non-negative definite function ∑ on T
T
T
×T, a Lévy measure ν on (ℝ , B ) and a function b ∈ ℝ such that for every
T
I ∈ T and a ∈ ℝ
I
̂

where ∑ is the restriction of ∑ to I x I. (∑, v, b) is called the generating triplet
I
of X. Conversely, given a generating triplet (∑, v, b) as above, there exists an
t t∈T satisfying (3).
infinitely divisible process X = (X )
∈ is a symmetric

By (3) one can decompose + , where = ( )
=
∈ is
Gaussian process with covariance function ∶ × ↦ ℝ and = ( )
an independent of G Poissonian process. By their independence, in many
situations, we can consider both processes separately. The Poissonian process
Y admits the following canonical spectral representation.
t t∈T be a Poissonian infinitely divisible process with
Proposition 3. Let Y = (Y )
Lévy measure ν and a shift function b. Let N be a Poisson random measure on
(ℝ , ) whose intensity measure equals to the Lévy measure ν of Y. Let χ be
T
T
̃
a (cutoff) function on ℝ satisfying⟦u⟧ = uχ(u), u ∈ ℝ. Then the process Y =
̃
(Y ) given by
t t∈T

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