Behavior of modes of a class of processes with independent increments
The process \(X=(X_ t,0\leq t<\infty)\) is said to belong to the class L iff X is a process with homogeneous independent increments and \(X_ t\) has the following characteristic function \[ E\{\exp (izX_ t)\}=\exp [t(i\gamma z-\sigma^ 2z^ 2+\int^{\infty}_{-\infty}g(z,x)x^{- 1}k(x)dx] \] where \(g(z,x)=e^{izx}-1-izx(1+x^ 2)^{-1}\), \(\gamma\) real, \(\sigma^ 2\geq 0\), k(x) is non-negative and non-increasing on (0,\(\infty)\), non-positive and non-increasing on (-\(\infty,0)\), and \[ \int_{| x| <1}xk(x)dx+\int_{| x| >1}x^{- 1}k(x)dx<\infty. \] According to a result of \textit{M. Yamazoto} [Ann. Probab. 6, 523-531 (1978; Zbl 0394.60017)], \(X_ t\) is unimodal for each t. Suppose that \(X_ t\) has a unique mode denoted by a(t). Thus we come to study the important problem of the behavior of a(t) as a function of t. The present paper is devoted to a solution of this problem. The author considers separately the case of increasing process X (subordinator) and the case when X is not an increasing process. In the first case the behavior of a(t) is found when k(x) is slowly varying at infinity while the second case is more difficult and here only some asymptotic results are obtained. The author has established a large number of results in a form of theorems and lemmas thus giving a nice and interesting description of the class L. Several related topics (e.g. concerning the stable processes) are discussed too.
scientific article; zbMATH DE number 3988402