Related papers: A fractional generalization of the Poisson process…
We consider a weighted sum of a series of independent Poisson random variables and show that it results in a new compound Poisson distribution which includes the Poisson distribution and Poisson distribution of order k. An explicit…
There exist important stochastic physical processes involving infinite mean waiting times. The mean divergence has dramatic consequences on the process dynamics. Fractal time random walks, a diffusion process, and subrecoil laser cooling, a…
This document presents a compilation of results related to the theory of stochastic processes, with a specific focus on Markov processes, regenerative processes, renewal processes, and stationary processes. The relevance of these topics…
Renewal processes are zero-dimensional processes defined by independent intervals of time between zero crossings of a random walker. We subject renewal processes them to stochastic resetting by setting the position of the random walker to…
Traditionally, fractional counting processes, such as the fractional Poisson process, etc. have been defined using fractional differential and integral operators. Recently, Laskin (2024) introduced a generalized fractional counting process…
The fractional Poisson process (FPP) generalizes the standard Poisson process by replacing exponentially distributed return times with Mittag-Leffler distributed ones with an extra tail parameter, allowing for greater flexibility. The FPP…
This paper introduces the Generalized Fractional Compound Poisson Process (GFCPP), which claims to be a unified fractional version of the compound Poisson process (CPP) that encompasses existing variations as special cases. We derive its…
Stochastic renewal processes are ubiquitous across physics, biology, and the social sciences. Here, we show that continuous-time renewal dynamics can naturally produce a mixed discrete-continuous structure, with a macroscopic fraction of…
We prove regenerative properties for the linear Hawkes process under minimal assumptions on the transfer function, which may have unbounded support. These results are applicable to sliding window statistical estimators. We exploit…
In many complex systems studied in statistical physics, inter-arrival times between events such as solar flares, trades and neuron voltages follow a heavy-tailed distribution. The set of event times is fractal-like, being dense in some time…
Time evolutions whose infinitesimal generator is a fractional time derivative arise generally in the long time limit. Such fractional time evolutions are considered here for random walks. An exact relationship is given between the…
This paper studies the joint moments of a compound discounted renewal process observed at different times with each arrival removed from the system after a random delay. This process can be used to describe the aggregate (discounted)…
We study the large-time asymptotic of renewal-reward processes with a heavy-tailed waiting time distribution. It is known that the heavy tail of the distribution produces an extremely slow dynamics, resulting in a singular large deviation…
We consider the drift and diffusion properties of periodically driven renewal processes. These processes are defined by a periodically time dependent waiting time distribution, which governs the interval between subsequent events. We show…
We present and establish large deviations principles for general multivariate renewal-reward processes associated with a classical discrete-time renewal process. A renewal-reward process describes a cumulative reward over time, supposing…
Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and its extensions for various counting processes. We first consider a…
In the paper we consider a generalizations of the notion of Poisson process to the case when classical convolution is replaced by generalized convolution in the sense of K. Urbanik [16] following two classical definitions of Poisson…
Renewal process is a point process where an inter-event time between successive renewals is an independent and identically distributed random variable. Alternating renewal process is a dichotomous process and a slight generalization of the…
The Poisson distribution is the probability distribution of the number of independent events in a given period of time. Although the Poisson distribution appears ubiquitously in various stochastic dynamics of gene expression, both as…
We analyze generalized space-time fractional motions on undirected networks and lattices. The continuous-time random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the…