In this article, we introduce an infinite-dimensional analogue of the $$\alpha $$ α -stable Lévy motion, defined as a Lévy process $$Z=\{Z(t)\}_{t \ge 0}$$ Z = { Z ( t ) } t ≥ 0 with values in the space $${\mathbb {D}}$$ D of càdlàg functions on [0, 1], equipped with Skorokhod’s $$J_1$$ J 1 topology. For each $$t \ge 0$$ t ≥ 0 , Z ( t ) is an $$\alpha $$ α -stable process with sample paths in $${\mathbb {D}}$$ D , denoted by $$\{Z(t,s)\}_{s\in [0,1]}$$ { Z ( t , s ) } s ∈ [ 0 , 1 ] . Intuitively, Z ( t , s ) gives the value of the process Z at time t and location s in space. This process is closely related to the concept of regular variation for random elements in $${\mathbb {D}}$$ D introduced in de Haan and Lin (Ann Probab 29:467–483, 2001 ) and Hult and Lindskog (Stoch Proc Appl 115:249–274, 2005 ). We give a construction of Z based on a Poisson random measure, and we show that Z has a modification whose sample paths are càdlàg functions on $$[0,\infty )$$ [ 0 , ∞ ) with values in $${\mathbb {D}}$$ D . Finally, we prove a functional limit theorem which identifies the distribution of this modification as the limit of the partial sum sequence $$\{S_n(t)=\sum _{i=1}^{[nt]}X_i\}_{t\ge 0}$$ { S n ( t ) = ∑ i = 1 [ n t ] X i } t ≥ 0 , suitably normalized and centered, associated with a sequence $$(X_i)_{i\ge 1}$$ ( X i ) i ≥ 1 of i.i.d. regularly varying elements in $${\mathbb {D}}$$ D .

中文翻译：

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在 Skorokhod 空间中具有值的稳定 Lévy 运动：构造和近似

在本文中，我们介绍了 $$\alpha $$ α 稳定 Lévy 运动的无限维模拟，定义为 Lévy 过程 $$Z=\{Z(t)\}_{t \ge 0}$ $Z = { Z ( t ) } t ≥ 0 值在空间 $${\mathbb {D}}$$ D 的 càdlàg 函数在 [0, 1] 上，配备了 Skorokhod 的 $$J_1$$ J 1 拓扑. 对于每个 $$t \ge 0$$ t ≥ 0 ，Z ( t ) 是 $$\alpha $$ α 稳定过程，样本路径在 $${\mathbb {D}}$$ D 中，用 $ 表示$\{Z(t,s)\}_{s\in [0,1]}$$ { Z ( t , s ) } s ∈ [ 0 , 1 ] 。直观地，Z ( t , s ) 给出了过程 Z 在时间 t 和空间位置 s 的值。这个过程与在 de Haan and Lin (Ann Probab 29:467–483, 2001) 和 Hult and Lindskog (Stoch Proc Appl 115:249–274, 2005)。我们基于泊松随机测度给出了 Z 的构造，并且我们证明了 Z 有一个修改，其样本路径是 $$[0,\infty )$$ [ 0 , ∞ ) 上的 càdlàg 函数，值在 $${\ mathbb {D}}$$ D . 最后，我们证明了一个函数极限定理，它将这个修改的分布标识为部分和序列的极限 $$\{S_n(t)=\sum_{i=1}^{[nt]}X_i\}_ {t\ge 0}$$ { S n ( t ) = ∑ i = 1 [ nt ] X i } t ≥ 0 ，适当归一化和居中，与序列 $$(X_i)_{i\ge 1} $$ ( X i ) i ≥ 1 个 iid 在 $${\mathbb {D}}$$ D 中定期变化的元素。