76 research outputs found
A review on symmetry properties of birth-death processes
In this paper we review some results on time-homogeneous birth-death
processes. Specifically, for truncated birth-death processes with two absorbing
or two reflecting endpoints, we recall the necessary and sufficient conditions
on the transition rates such that the transition probabilities satisfy a
spatial symmetry relation. The latter leads to simple expressions for
first-passage-time densities and avoiding transition probabilities. This
approach is thus thoroughly extended to the case of bilateral birth-death
processes, even in the presence of catastrophes, and to the case of a
two-dimensional birth-death process with constant rates.Comment: 16 pages, 4 figure
On the Effect of Random Alternating Perturbations on Hazard Rates
We consider a model for systems perturbed by dichotomous noise, in which the
hazard rate function of a random lifetime is subject to additive
time-alternating perturbations described by the telegraph process. This leads
us to define a real-valued continuous-time stochastic process of alternating
type expressed in terms of the integrated telegraph process for which we obtain
the probability distribution, mean and variance. An application to survival
analysis and reliability data sets based on confidence bands for estimated
hazard rate functions is also provided.Comment: 14 pages, 6 figure
On a bilateral birth-death process with alternating rates
We consider a bilateral birth-death process characterized by a constant
transition rate from even states and a possibly different transition
rate from odd states. We determine the probability generating functions
of the even and odd states, the transition probabilities, mean and variance of
the process for arbitrary initial state. Some features of the birth-death
process confined to the non-negative integers by a reflecting boundary in the
zero-state are also analyzed. In particular, making use of a Laplace transform
approach we obtain a series form of the transition probability from state 1 to
the zero-state.Comment: 13 pages, 3 figure
Compound Poisson process with a Poisson subordinator
A compound Poisson process whose randomized time is an independent Poisson
process is called compound Poisson process with Poisson subordinator. We
provide its probability distribution, which is expressed in terms of the Bell
polynomials, and investigate in detail both the special cases in which the
compound Poisson process has exponential jumps and normal jumps. Then for the
iterated Poisson process we discuss some properties and provide convergence
results to a Poisson process. The first-crossing-time problem for the iterated
Poisson process is finally tackled in the cases of (i) a decreasing and
constant boundary, where we provide some closed-form results, and (ii) a
linearly increasing boundary, where we propose an iterative procedure to
compute the first-crossing-time density and survival functions.Comment: 16 pages, 7 figure
A multispecies birth-death-immigration process and its diffusion approximation
We consider an extended birth-death-immigration process defined on a lattice
formed by the integers of semiaxes joined at the origin. When the process
reaches the origin, then it may jumps toward any semiaxis with the same rate.
The dynamics on each ray evolves according to a one-dimensional linear
birth-death process with immigration. We investigate the transient and
asymptotic behavior of the process via its probability generating function. The
stationary distribution, when existing, is a zero-modified negative binomial
distribution. We also study a diffusive approximation of the process, which
involves a diffusion process with linear drift and infinitesimal variance on
each ray. It possesses a gamma-type transient density admitting a stationary
limit.
As a byproduct of our study, we obtain a closed form of the number of
permutations with a fixed number of components, and a new series form of the
polylogarithm function expressed in terms of the Gauss hypergeometric function.Comment: 26 pages, 7 figure
On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities
We consider a bilateral birth-death process having sigmoidal-type rates. A thorough discussion on its transient behaviour is given, which includes studying symmetry properties of the transition probabilities, finding conditions leading to their bimodality, determining mean and variance of the process, and analyzing absorption problems in the presence of 1 or 2 boundaries. In particular, thanks to the symmetry properties we obtain the avoiding transition probabilities in the presence of a pair of absorbing boundaries, expressed as a series
Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates
We consider continuous-time Markov chains on integers which allow transitions
to adjacent states only, with alternating rates. We give explicit formulas for
probability generating functions, and also for means, variances and state
probabilities of the random variables of the process. Moreover we study
independent random time-changes with the inverse of the stable subordinator,
the stable subordinator and the tempered stable subodinator. We also present
some asymptotic results in the fashion of large deviations. These results give
some generalizations of those presented in Di Crescenzo A., Macci C.,
Martinucci B. (2014).Comment: 25 pages, 2 figure
A Quantile-Based Probabilistic Mean Value Theorem
For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.The research of A.D.C. and B.M. has been performed under partial support by GNCS-INdAM and Regione Campania (Legge 5). J.M. was supported by project MTM2012-34023-FEDER, “Comparación y dependencia en modelos probabilísticos con aplicaciones en fiabilidad y riesgos”, from Universidad de Murcia
Stochastic Processes with Applications
Stochastic processes have wide relevance in mathematics both for theoretical aspects and for their numerous real-world applications in various domains. They represent a very active research field which is attracting the growing interest of scientists from a range of disciplines.This Special Issue aims to present a collection of current contributions concerning various topics related to stochastic processes and their applications. In particular, the focus here is on applications of stochastic processes as models of dynamic phenomena in research areas certain to be of interest, such as economics, statistical physics, queuing theory, biology, theoretical neurobiology, and reliability theory. Various contributions dealing with theoretical issues on stochastic processes are also included
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