664 research outputs found
On Stein's Method for Infinitely Divisible Laws With Finite First Moment
We present, in a unified way, a Stein methodology for infinitely divisible
laws (without Gaussian component) having finite first moment. Based on a
correlation representation, we obtain a characterizing non-local Stein operator
which boils down to classical Stein operators in specific examples. Thanks to
this characterizing operator, we introduce various extensions of size bias and
zero bias distributions and prove that these notions are closely linked to
infinite divisibility. Combined with standard Fourier techniques, these
extensions also allow obtaining explicit rates of convergence for compound
Poisson approximation in particular towards the symmetric -stable
distribution. Finally, in the setting of non-degenerate self-decomposable laws,
by semigroup techniques, we solve the Stein equation induced by the
characterizing non-local Stein operator and obtain quantitative bounds in weak
limit theorems for sums of independent random variables going back to the work
of Khintchine and L\'evy.Comment: 58 pages. Minor changes and new results in Sections 5 and
On Layered Stable Processes
Layered stable (multivariate) distributions and processes are defined and
studied. A layered stable process combines stable trends of two different
indices, one of them possibly Gaussian. More precisely, in short time, it is
close to a stable process while, in long time, it approximates another stable
(possibly Gaussian) process. We also investigate the absolute continuity of a
layered stable process with respect to its short time limiting stable process.
A series representation of layered stable processes is derived, giving insights
into both the structure of the sample paths and of the short and long time
behaviors. This series is further used for sample paths simulation.Comment: 22 pages, 9 figure
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