A generalization result regarding the small and large scale behavior of infinitely divisible processes
Rather general conditions for the convergence of normed time-scaled stochastic integrals of independently scattered L\(\acute{e}\)vy random measures to a limit are considered. The aim of the paper is to generalize previous results to the greater class of stochastically continuous infinitely divisible processes with no Gaussian part and expand the results to the multi-dimensional case. Let \(X_t\) be a stochastic process \[ X_t: = \int_S f_t(s) \left( N(ds) - \frac{\nu (ds)}{1 \vee \parallel f_t(s) \parallel} \right), \] where \({N}\) is a Poisson random measure on a measurable space (S, s) with intensity \(\nu\) and \(\{f_t(s)\}\) are measurable deterministic functions in \(\mathbb{R}^d\). In the paper, the convergence of properly normed and centered time-scaled process \(X_{\beta t}\) to a stable or Gaussian limit as \(\beta \rightarrow \beta_0\) is considered. In other words, the aim of the paper is to determine rather wide conditions on norming sequences \(n_\beta\) and centers \(c_{\beta, t}\) in order for the processes \(n_\beta X_{\beta t} - c_{\beta, t}\) to converge in the sense of finite-dimensional distributions (\(\overset{\mathrm{fdd}}{}\)) as the scaling factor \(\beta\) tends to some value \(\beta_0\). Both small (\(\beta_0 = 0\)) and large (\(\beta_0 = \infty\)) scaling factors are of interest. The main result is Theorem 3.1. Under the conditions that there exist one-to-one invertible measurable maps \(\phi_\beta\) from \(S\) into \(S\) with \(\phi^{-1}_\beta\) measurable and normalized sequences \(n_\beta\) and \(m_\beta\) satisfying \(\text{a.e.-}\nu\), for any \(t \in \mathbb{R}\) and for some functions \(k_t\), \(h_t\), \(g\) and \(l\) on the space \(S\), \[ m_\beta f_{\beta t} (\phi_\beta(s)) \rightarrow h_t(s); \quad \parallel m_\beta f_{\beta t} (\phi_\beta(s))\parallel \leq k_t(s); \] \[ \frac{d(\nu \circ \phi_\beta)}{d \nu} (s) \left( \frac{n_\beta}{m_\beta} \right)^2 \rightarrow g(s); \quad \frac{d(\nu \circ \phi_\beta)}{d \nu} (s) \left( \frac{n_\beta}{m_\beta} \right)^2 \leq l(s) \] as \(\beta \rightarrow \beta_0\). Let \[ c_{\beta, t} = \int_S \left( \frac{n_\beta f_{\beta t}(s)}{1 \vee \parallel n_\beta f_{\beta t}(s) \parallel} - \frac{n_\beta f_{\beta t}(s)}{1 \vee \parallel f_{\beta t}(s) \parallel} \right) \nu (ds). \] Then \(n_\beta X_{\beta t} - c_{\beta, t} \overset{\mathrm{fdd}}{} Y_t\) as \(\beta \rightarrow \beta_0\), where the process \(Y_t\) satisfies on of the two cases (i) if \(\frac{n_\beta}{m_\beta} \rightarrow 0\) and \(l \cdot k_t^2 \in L_\nu(S)\), then \(Y_t = \int_S h_t(s) Z(ds)\), where \(Z\) is an independently scattered Gaussian random measure with control measure \((g \cdot \nu)(ds) := g(s) \nu(ds)\); or (ii) if \(m_\beta = n_\beta\) and \(l \cdot \{1 \vee k^2_t\} \in L_\nu(S)\), then \[ Y_t = \int_S h_t(s) \left( M(ds) - \frac{g(s) \nu(ds)}{1 \vee \| h_t(s) \|} \right), \] where \({M}\) is a Poisson random measure with intensity \((g \cdot \nu)(ds) := g(s) \nu(ds)\). The main theorem is proved in Section 3. Section 2 contains lemmas needed for the proof. Section 4 presents corollaries with proofs and their extensions to the multi-dimensional case. Corollary 4.1. deals with processes \[ X_t = \int_S f_t(s) ( N(ds) - \nu(ds)). \] Corollary 4.2 is a new statement for processes given by \[ X_t = \int_S f_t(s) N(ds). \] In Section 5, the stable Lévy motions, tempered Lévy motions, standard Poisson telecom process and Lamperti stable processes are considered as examples.
scientific article; zbMATH DE number 6112310