Compositions of mappings of infinitely divisible distributions with applications to finding the limits of some nested subclasses

Abstract

Let $\{X_t^{(\mu)},t\ge 0\}$ be a L\'evy process on $R^d$ whose distribution at time 1 is $\mu$, and let $f$ be a nonrandom measurable function on $(0, a), 0 < a\leq \infty$. Then we can define a mapping $\Phi_f(\mu)$ by the law of $\int_0^af(t)dX_t^{(\mu)}$, from $\mathfrak D(\Phi_f)$ which is the totality of $\mu\in I(R^d)$ such that the stochastic integral $\int_0^af(t)dX_t^{(\mu)}$ is definable, into a class of infinitely divisible distributions. For $m\in N$, let $\Phi_f^m$ be the $m$ times composition of $\Phi_f$ itself. Maejima and Sato (2009) proved that the limits $\bigcap_{m=1}^\infty\Phi^m_f(\mathfrak D(\Phi^m_f))$ are the same for several known $f$'s. Maejima and Nakahara (2009) introduced more general $f$'s. In this paper, the limits $\bigcap_{m=1}^\infty\Phi^m_f(\mathfrak D(\Phi^m_f))$ for such general $f$'s are investigated by using the idea of compositions of suitable mappings of infinitely divisible distributions.