Continuity and Boundedness of Infinitely Divisible Processes: A Poisson Point Process Approach
Continuity and Boundedness of Infinitely Divisible Processes: A Poisson Point Process Approach
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无限可分过程的连续性和有界性：泊松点过程方法
DOI：
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发表时间：
2005
期刊：
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通讯作者：
J. Rosínski
中科院分区：
文献类型：
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作者：
M. Marcus;J. Rosínski
获得了随机连续无限可分过程的有界性和连续性的充分条件，没有高斯分量，{Y(t),t ∈T}，其中T是紧度量空间或伪度量空间。这样的过程有一个由 Y(t)=X(t)+b(t),t∈T 给出的版本，其中 b 是确定性漂移函数，
$$X(t) = int_S f(t,s) left[N(ds)-(|f(t,s)|vee 1)^{-1}
u(ds)
ight].$$ 这里 N 是 Borel 空间 S 上的泊松随机测度，具有 σ−有限平均测度 ν，并且
$$f{:} 时间 S 映射到 R$$ 是一个可测量的确定性函数。令 τ: T2 → R+ 为 T 上的连续伪度量。定义 f 各部分的 τ-Lipschitz 范数为
$$ |f|_{ au} (s)=D^{-1}f(t_0,s) + sulimits_{u,vin T} {{|f(u,s)-f(v,s)|}over{ au(u,v)}}$$ 对于某些 t0 ∈T，其中 D 是 (T,τ) 的直径。 X 有界性和连续性的充分条件以测度形式给出
$$
u,||f||_ au$$ 并主要由 τ 确定的度量和/或度量熵条件。它们应用于以下形式的随机积分
$$Y(t) = int_S g(t,s)M(ds)quadtin T,$$ 其中 M 是零均值、独立分散、无限可分的随机测量，没有高斯分量。给出的几个例子表明，在许多情况下获得的条件是相当尖锐的。除了获得连续性和有界性的条件之外，还获得了弱 Lp 范数和强 Lp 范数的界
$$sup_{锡 T}|X(t)|$$ 和
$$sup_{ au (t,u)leqslantdelta,t,u in T}|X(t) - X(u)|$$ 对于所有
$$0<delta leqslant D$$。这些结果取决于矩的不等式以及弱函数和强函数的相关函数
$$ell^p$$ 序列 {xj} 的范数，它们是 R+ 上泊松点过程 M 的事件，并以 M 的强度度量形式给出。这些结果具有独立的意义。
Sufficient conditions for boundedness and continuity are obtained for stochastically continuous infinitely divisible processes, without Gaussian component, {Y(t),t ∈T}, where T is a compact metric space or pseudo-metric space. Such processes have a version given by Y(t)=X(t)+b(t),t∈T where b is a deterministic drift function and
$$X(t) = int_S f(t,s) left[N(ds)-(|f(t,s)|vee 1)^{-1}
u(ds)
ight].$$ Here N is a Poisson random measure on a Borel space S with σ−finite mean measure ν, and
$$f{:} T imes S mapsto R$$ is a measurable deterministic function. Let τ: T2 → R+ be a continuous pseudo–metric on T. Define the τ-Lipschitz norm of the sections of f by
$$ |f|_{ au} (s)=D^{-1}f(t_0,s) + suplimits_{u,vin T} {{|f(u,s)-f(v,s)|}over{ au(u,v)}}$$ for some t0 ∈T, where D is the diameter of (T,τ). The sufficient conditions for boundedness and continuity of X are given in terms of the measure
$$
u,||f||_ au$$ and majorizing measure and or metric entropy conditions determined by τ. They are applied to stochastic integrals of the form
$$Y(t) = int_S g(t,s)M(ds)quad tin T,$$ where M is a zero-mean, independently scattered, infinitely divisible random measure without Gaussian component. Several examples are given which show that in many cases the conditions obtained are quite sharp. In addition to obtaining conditions for continuity and boundedness, bounds are obtained for the weak and strong Lp norms of
$$sup_{tin T}|X(t)|$$ and
$$sup_{ au (t,u)leqslantdelta,t,u in T}|X(t) - X(u)|$$ for all
$$0<delta leqslant D$$. These results depend on inequalities for moments and related functions of the weak and strong
$$ell^p$$ norms of sequences {xj}, which are the events of Poisson point process M on R+ and are given in terms of the intensity measure of M. These results are of independent interest.