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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...
We introduce and study a class of random walks defined on the integer lattice -a discrete space and time counterpart of the symmetric α-stable process in . When 0 < α <2 any coordinate axis in , 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.
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 equations and, when Y is a stable subordinator, can arise...
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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