Symmetric Stable Process Of Index Research Articles

Polar Sets and Multiple Points for Super-Brownian Motion

We study the closed support of the measure-valued diffusions of Watanabe and Dawson. When the spatial motion is Brownian, sufficient conditions involving capacity are given for a fixed set to be hit by the $k$-multiple points of the support process. The conditions are close to the necessary conditions found by Dawson, Iscoe and Perkins and lead to necessary and sufficient conditions for the existence of $k$-multiple points. When the spatial motion is a symmetric stable process of index $\alpha < 2$, the closed support is shown to be $\mathbb{R}^d$ or $\varnothing$.

Dyadic approximation of double integrals with respect to symmetric stable processes

Recently, Rosinski and Woyczynski have given necessary and sufficient conditions for the existence of the double integral with respect to a symmetric stable process of index α in [1, 2). In their approach the double integral is defined as an iterated Itô-type integral. We show here that it can also be defined as the limit of integrals of step functions and that the two approaches are equivalent. For many purposes this result reduces the study of double integrals to that of quadratic forms in independent stable random variables.

Laws of the iterated logarithm for symmetric stable processes

Let X(t) be a symmetric stable process of index α> 1 with local time L(t, x) and define R(t)=¦{x:X(s)=x for some s≦t}¦ and L *(t) = sup{L(t, x): x∈ℝ1}. We prove that $$\begin{gathered} \mathop {\lim \sup }\limits_{t \to \infty } t^{ - 1/\alpha } (\log \log {\text{t}})^{ - (1 - 1/\alpha )} R(t) = c_1 {\text{ a}}{\text{.s}}{\text{.}} \hfill \mathop {\lim \inf }\limits_{t \to \infty } t^{ - (1 - 1/\alpha )} (\log \log {\text{t}})^{1 - 1/\alpha } L*(t) = c_2 {\text{ a}}{\text{.s}}{\text{.}} \hfill \end{gathered} $$ where c 1, c 2∈(0, ∞). A law of the iterated logarithm is also given for X(t) when its large jumps are excluded.

On the Paths of Symmetric Stable Processes

It is shown that if $X(t), t \geqslant 0$, is a symmetric stable process of index $\alpha , 0 < \alpha < 2$, then $\sup _t \lim \inf _{h \downarrow 0} (X(t + h) - X(t))h^{-1/\alpha } = \infty$ a.s. This settles a question of Fristedt about strictly stable subordinators.

On the spectral representation of symmetric stable processes

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L p (0 < p ≤ 2). We use the theory of L p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L q into real or complex L p for 0 < p < q ≤ 2.

The Carrying Dimension of a Stochastic Measure Diffusion

A multiplicative stochastic measure diffusion process in $R^d$ is the continuous analogue of an infinite particle branching Markov process in which the particles move in $R^d$ according to a symmetric stable process of index $\alpha 0 \alpha$. The latter result is also proved by a different approach in the case $d = \alpha$.

Using additive functionals to embed preassigned distributions in symmetric stable processes

Following Skorokhod, several authors in recent years have proposed methods to define a stopping time $T$ for Brownian motion $({X_t},{\mathcal {F}_t})$ such that ${X_T}$ will have some preassigned distribution. In this paper a method utilizing additive functionals is explored. It is applicable not only to Brownian motion but all symmetric stable processes of index $\alpha > 1$. Using this method one is able to obtain any distribution having a finite $\alpha - 1$ absolute moment. There is also a discussion of the problem of approximating symmetric stable processes with random walks.

Using Additive Functionals to Embed Preassigned Distributions in Symmetric Stable Processes

Following Skorokhod, several authors in recent years have proposed methods to define a stopping time $T$ for Brownian motion $({X_t},{\mathcal {F}_t})$ such that ${X_T}$ will have some preassigned distribution. In this paper a method utilizing additive functionals is explored. It is applicable not only to Brownian motion but all symmetric stable processes of index $\alpha > 1$. Using this method one is able to obtain any distribution having a finite $\alpha - 1$ absolute moment. There is also a discussion of the problem of approximating symmetric stable processes with random walks.

The variation of a stable path is stable

Let X(t) be a separable symmetric stable process of index α. Let P be a finite partition of [0,1], and ℘ a collection of partitions. The variation of a path X(t) is defined in three ways in terms of the sum \(\sum\limits_{t_i \in P} {|X(t_i ) - X(t_{i - 1} )|^\beta } \) collection ℘. Under certain conditions on ℘ and on the parameters α and Β, the distribution of the variation is shown to be a stable law. Under other conditions the distribution of the variational sum converges to a stable distribution.