On the notion of multiple Markov S\(\alpha\)S processes (Q1894030)

Let \(X(t)\), \(t \in R\), be a stochastic process which is not necessarily Gaussian nor 2nd order. \(X(t)\) is called \(n\)-ple Markov of LC type if there exists nontrivial coefficients \(a_j\) such that \[ P \left( \sum^n_{j = 1} a_jX(t_j) \in B \mid {\mathcal B}_{t_0}(X) \right) = P \left( \sum a_j X(t_j) \in B \mid X (t_0) \right), \] for any \(B \in {\mathcal B} (R)\) and \(t_i \in R\), and other two additional conditions. It is easy to see that an 1-ple Markov process of LC type is a Markov process and that in the Gaussian case this Markov property coincides with that of T. Hida. Let \(Z(t)\), \(t \in R\), be a symmetric \(\alpha\) stable (S\(\alpha\)S) process with independent increment. The integral expression of a process \(X(t) = \int^t F(t,u) dZ(u)\) is called canonical if two \(\sigma\) fields coincide, \({\mathcal B}_t (X) = {\mathcal B}_t (Z)\), and is called proper if two linear spaces coincide, \({\mathcal M}_t (X) = {\mathcal M}_t (Z)\). The author obtains an extension of the Hida-Cramer's theorem to S\(\alpha\)S case. Assume that an S\(\alpha\)S process \(X(t)\) has a canonical representation, then \(X\) is \(n\)-ple Markov of LC type if and only if the kernel function \(F(t,u)\) is a Goursat kernel of order \(n\).