Subordination (in the sense of Bochner) is a random time-change of a stochastic process \(X(t)\) w.r.t. a stochastically independent \(\mathbb{R}^+\)-valued increasing Lévy process \(T(t)\). Such processes are often called subordinators and the time-changed process \(X(T(t))\) is said to be subordinate to \(X(t)\). If \(X(t)\) is itself a Lévy process, so is the subordinate process. Subordination was first studied by S. Bochner [Ann. Math., II. Ser. 48, 1014-1061 (1947; Zbl 0029.36802)] and since then many authors studied various aspects of this technique; see J. Bertoin [in: Lectures on probability theory and statistics. Lect. Notes Math. 1717, 1-91 (1999; Zbl 0955.60046)] for a survey.
Let \(X_j(t)\), \(1\leq j\leq d\), be \(d\) independent \(\mathbb{R}^{n_j}\)-valued Lévy processes and arrange them into a single column vector \[ \underline s= (s_1,\dots, s_d)\mapsto Y(\underline s)={^t}(X_1(s_1),\dots, X_d(s_d)) \] which is interpreted as a \(\sum_j n_j\)-dimensional multiparameter process indexed by \(\mathbb{R}^d_+\). A \(d\)-dimensional subordinator is an independent \(\mathbb{R}^d_+\)-valued process \(\underline T(t)\) whose coordinate processes are subordinators; the process \(\mathbb{R}^+\ni t\mapsto Y(\underline T(t))\) is called subordinate process.
The authors briefly discuss some methods how to construct (multivariate) subordinators \(\underline T(t)\) and give some examples; the Lévy-Khinchin formula for \(\underline T(t)\) is taken from A. V. Skorokhod [“Random processes with independent increments” (1991; Zbl 0732.60081), §3.21], see also S. Bochner [“Harmonic analysis and the theory of probability” (1955; Zbl 0068.11702), §4.7], where a closely related concept of subordination in several variables is discussed. As an application of the Lévy-Khinchin decomposition, the Lévy triplet of the subordinate process is calculated. The proof follows closely the classical (i.e., one-parameter) case.
In a further section, the notion of multiparameter Lévy processes is introduced; essentially, the linear order \(\leq\) on \(\mathbb{R}^+\) is now being replaced by the partial order on \(\mathbb{R}^d_+\), where \(\underline s\preceq\underline t\) if for each coordinate \(s_j\leq t_j\). This allows to extend the defining properties of a Lévy process to the multiparameter setting. The main result is that a subordinate multiparameter Lévy process is a (classical one-parameter) Lévy process whose Lévy triplet can be explicitly calculated. The proof of this theorem follows also classical lines, examining the characteristic function (Fourier transform) of the transition function. The technique of multivariate subordination is now applied to questions of operator stability and operator self-decomposability. Recall that random vector is operator self-decomposable if \(X= b^{-Q}X+ Y_b\) (in law) for all \(b> 1\), and strictly operator stable, if for each \(n\in\mathbb{N}\) and \(n\) independent copies \(X_j\) of \(X\) we have \(X_1+\cdots+ X_n= n^Q X\) (in law); \(Q\) is a fixed matrix.
Here is the main result: If each of the Lévy processes \(X_j(t)\) is \(Q_j\) strictly operator stable, and if \(\underline T(t)\) is \(\text{diag}(h_1,\dots, h_d)\) self-decomposable, then the subordinate process is \(\text{diag}(h_1Q_1,\dots, h_dQ_d)\) self-decomposable. The proof of this result uses a characterization of self-decomposability due to K.-i. Sato and M. Yamazato [Nagoya Math. J. 97, 71-94 (1985; Zbl 0577.60025)]. Similarly, the subordinate process inherits the operator version of the class \(L_m\)-property in the sense of Urbanik and Sato. Finally, an application to random vectors and processes of (generalized) type “\(G\)” (“\(G(Q)\)”) is given which turn out to be special cases of multivariate subordination.