COMPOSITIONS OF MAPPINGS OF INFINITELY DIVISIBLE DISTRIBUTIONS WITH APPLICATIONS TO FINDING THE LIMITS OF SOME NESTED SUBCLASSES

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.

As the definition of stochastic integrals, we adopt the method in Sato [25,26]. It is known that if f is locally square integrable on [0, ∞), then t 0 f (s)d X s , t ∈ [0, ∞), is definable for any Lévy process {X t }. The improper stochastic integral ∞ 0 f (s)d X s is defined as the limit in probability of t 0 f (s)dX s as t → ∞ whenever the limit exists. In our definition, { t 0 f (s)dX s , t ∈ [0, ∞)} is an additive process in law, which is not always càdlàg in t. If we take its càdlàg modification, the convergence of t 0 f (s)dX s above is equivalent to the almost sure convergence of the modification as t → ∞. Let {X (µ) t , t ≥ 0} stand for a Lévy process on d with (X (µ) 1 ) = µ. Using this Lévy process, we can define a mapping is called completely closed in the strong sense (abbreviated as c.c.s.s.) if H is closed under type equivalence, convolution, weak convergence and t-th convolution for any t > 0. We list below several known mappings. In the following, I log ( d ) denotes the totality of µ ∈ I( d ) satisfying d log + |x|µ(d x) < ∞, where log + |x| = (log |x|) ∨ 0. (2) Φ-mapping (Wolfe [30], Jurek and Vervaat [15], Sato and Yamazato [29]): Let and let L( d ) be the class of selfdecomposable distributions on d . Then L( d ) = Φ(I log ( d )).
We call M ( d ) := (I log ( d )) the class M and it was actually introduced in Aoyama et al. [3] in the symmetric case.

Remark 1.1.
Jurek [13] introduced the mapping which is the same as Υ in (1.2) by the time change of the driving Lévy process. In the same way, it holds that Using this type of time change, we might avoid taking inverse functions as integrands of stochastic integral mappings. However, recently in Sato [27], Barndorff-Nielsen et al. [5] and other papers, they have used stochastic integral mappings whose integrands are some inverse functions and driving Lévy processes have original time parameter. In this paper, we also use this type of expressions.
Hereafter we denote the closure under weak convergence and convolution of a class [24] or Rocha-Arteaga and Sato [23], this is proved via the following fact: where Γ is a measure on (0, 2) satisfying and λ α is a probability measure on S := {ξ ∈ d : |ξ| = 1} for each α ∈ (0, 2), and λ α (C) is measurable in α ∈ (0, 2) for every C ∈ (S). This Γ is uniquely determined by µ and this λ α is uniquely determined by µ up to α of Γ-measure 0. For the case in more general spaces, see Jurek [7]. For a set A ∈ ((0, 2)), let L A . In Maejima and Sato [18], nested subclasses R( m ), R(Υ m ), R(Ψ m ) and R( m ), m ∈ , were studied and the limits of these nested subclasses were proved to be equal to S( d ), (see also Jurek [12]). Furthermore, Sato [28] proved that the mappings Ψ α,1 , α ∈ (0, 2) produce smaller classes than S( d ) as the limit of iteration. Maejima and Ueda [19] showed that the mapping Φ α has the same iterated limit as that of Ψ α,1 for α ∈ (0, 2). Maejima and Ueda [20] also constructed a mapping producing a larger class than S( d ), which is the closure of the class of semi-stable distributions with a fixed span. The purpose of this paper is to find the limit of the nested subclasses R(Ψ m α,β ), m ∈ . For that, we start with the composition of Ψ α−β,β and Φ α , which will be used for characterizing the nested subclasses R(Ψ m α,β ), m ∈ .

Results
For β > 0, let The following lemma is trivial. Here The following result on composition will be a key in the proof of the main theorem, Theorem 2.4.
We also have the following.

Proofs
We first prove Theorem 2.2.
Proof of Theorem 2.2. For µ ∈ I( d ), we have and Propositions 3.4 and 2.17 of Sato [26] yields the finiteness of (3.1). Then we can use Fubini's theorem and have by a similar calculation to (3.1). This yields that . Note that the domains D(Ψ α,β ) and D(Φ α ) are the same and decreasing in α < 2 with respect to set inclusion due to Remark to Theorem 2.8 of Sato [27].
We finally prove Theorem 2.6.