282 research outputs found
On the infinite divisibility of distributions of some inverse subordinators
We consider the infinite divisibility of distributions of some well-known
inverse subordinators. Using a tail probability bound, we establish that
distributions of many of the inverse subordinators used in the literature are
not infinitely divisible. We further show that the distribution of a renewal
process time-changed by an inverse stable subordinator is not infinitely
divisible, which in particular implies that the distribution of the fractional
Poisson process is not infinitely divisible.Comment: Published at https://doi.org/10.15559/18-VMSTA108 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
Delayed and rushed motions through time change
We introduce a definition of delayed and rushed processes in terms of
lifetimes of base processes and time-changed base processes. Then, we consider
time changes given by subordinators and their inverse processes. Our analysis
shows that, quite surprisingly, time-changing with inverse subordinators does
not necessarily imply delay of the base process. Moreover, time-changing with
subordinators does not necessarily imply rushed base process.Comment: to appear on ALEA - Latin American Journal of Probability and
Mathematical Statistic
On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators
In the paper we present the governing equations for marginal distributions of
Poisson and Skellam processes time-changed by inverse subordinators. The
equations are given in terms of convolution-type derivatives
Convolution-type derivatives, hitting-times of subordinators and time-changed -semigroups
In this paper we will take under consideration subordinators and their
inverse processes (hitting-times). We will present in general the governing
equations of such processes by means of convolution-type integro-differential
operators similar to the fractional derivatives. Furthermore we will discuss
the concept of time-changed -semigroup in case the time-change is
performed by means of the hitting-time of a subordinator. We will show that
such time-change give rise to bounded linear operators not preserving the
semigroup property and we will present their governing equations by using again
integro-differential operators. Such operators are non-local and therefore we
will investigate the presence of long-range dependence.Comment: Final version, Potential analysis, 201
Time-Changed Poisson Processes
We consider time-changed Poisson processes, and derive the governing
difference-differential equations (DDE) these processes. In particular, we
consider the time-changed Poisson processes where the the time-change is
inverse Gaussian, or its hitting time process, and discuss the governing DDE's.
The stable subordinator, inverse stable subordinator and their iterated
versions are also considered as time-changes. DDE's corresponding to
probability mass functions of these time-changed processes are obtained.
Finally, we obtain a new governing partial differential equation for the
tempered stable subordinator of index when is a rational
number. We then use this result to obtain the governing DDE for the mass
function of Poisson process time-changed by tempered stable subordinator. Our
results extend and complement the results in Baeumer et al. \cite{B-M-N} and
Beghin et al. \cite{BO-1} in several directions.Comment: 18 page
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