Self-similar processes with independent increments associated with Lévy and Bessel processes.
A random variable \(X_1\) with a self-decomposable distribution, i.e. \(X_1\overset \text{d} = uX_1+\widehat X_u\) (in distribution) for each \(0<u<1\), can be represented by means of some Lévy process \(Y=(Y_s,s\geq 0)\) with \({\mathbf E}[\log (1\lor | Y_s| )]<\infty \) for all \(s\), the so-called background driving Lévy process of \(X_1\) [\textit{S.\ J.\ Wolfe}, Stochastic Processes Appl. 12, 301--312 (1982; Zbl 0482.60062)]. On the other hand, \textit{K. Sato} [Probab. Theory Relat. Fields 89, 285--300 (1991; Zbl 0725.60034)] showed that a distribution is self-decomposable iff for any fixed \(H>0\) it is the distribution of \(X_1\) for some additive \(H\)-self-similar process \((X_r)_{r\geq 0}\) (i.e.\ \((X_{cr})_{r\geq 0}\overset \text{d} = (c^HX_r)_{r\geq 0}\)). It is shown that either of these representations follow from the other. The equivalence theorem is proved, the relation to the Ornstein-Uhlenbeck process is mentioned. Finally, an alternative representation of the background driving Lévy process is given for a recurrent Bessel process (\(R_t\), \(t\geq 0\)) with positive real dimension \(\delta (1+\nu )\) and index \(\nu \in (-1,0)\), started at \(R_0=0\).
scientific article; zbMATH DE number 2138925
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