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α-stable limits for multiple channel queues in heavy traffic

Zbigniew Michna (2003)

Applicationes Mathematicae

We consider a sequence of renewal processes constructed from a sequence of random variables belonging to the domain of attraction of a stable law (1 < α < 2). We show that this sequence is not tight in the Skorokhod J₁ topology but the convergence of some functionals of it is derived. Using the structure of the sample paths of the renewal process we derive the convergence in the Skorokhod M₁ topology to an α-stable Lévy motion. This example leads to a weaker notion of weak convergence. As...

α-stable random walk has massive thorns

Alexander Bendikov, Wojciech Cygan (2015)

Colloquium Mathematicae

We introduce and study a class of random walks defined on the integer lattice ℤ d -a discrete space and time counterpart of the symmetric α-stable process in ℝ d . When 0 < α <2 any coordinate axis in ℤ d , d ≥ 3, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for a thorn to be a massive set.

α-time fractional brownian motion: PDE connections and local times

Erkan Nane, Dongsheng Wu, Yimin Xiao (2012)

ESAIM: Probability and Statistics

For 0 &lt; α ≤ 2 and 0 &lt; H &lt; 1, an α-time fractional Brownian motion is an iterated process Z =  {Z(t) = W(Y(t)), t ≥ 0}  obtained by taking a fractional Brownian motion  {W(t), t ∈ ℝ} with Hurst index 0 &lt; H &lt; 1 and replacing the time parameter with a strictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when Y is a stable subordinator, can arise...

α-time fractional Brownian motion: PDE connections and local times∗

Erkan Nane, Dongsheng Wu, Yimin Xiao (2012)

ESAIM: Probability and Statistics

For 0 < α ≤ 2 and 0 < H < 1, an α-time fractional Brownian motion is an iterated process Z =  {Z(t) = W(Y(t)), t ≥ 0}  obtained by taking a fractional Brownian motion  {W(t), t ∈ ℝ} with Hurst index 0 < H < 1 and replacing the time parameter with a strictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that such processes have natural connections to partial differential...

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