Comparison and Deviation from a Representation Formula

Let X ~ ID(b, ΣE, v), i.e., let X be a d-dimensional infinitely divisible random vector with characteristic function (1.1)$$\varphi (t) = \exp \left\{ {i\langle t,b\rangle - \frac{1}{2}\langle \sum t ,t\rangle + \int_{{\mathbb{R}^d}} {({e^{i\langle t,u\rangle }} - 1 - i\langle t,u\rangle 1(\left| u \right|} < 1))\nu (du)} \right\},$$ where t, b ∈ ℝd, Σ is a positive semidefinite d × d matrix and v (the Lévy measure) is a positive measure on B(ℝd), the Borel σ-algebra of ℝd, without atom at the origin and such that $$\int_{\mathbb{R}^d } {(\left| u \right|} ^2 \wedge 1)\nu (du) < + \infty (\langle \cdot , \cdot \rangle {e_{1}}, \ldots ,{e_{n}},e_{i}^{2} = - 1, $$ and ∣ · ∣ are respectively the Euclidean inner product and norm in ℝd).