arXiv subjects

M. Dhillon

Publications and source records attributed to M. Dhillon.

4 recordsLinked to original sources

arXiv2025

On Elephant Random Walk with Random Memory

In this paper, we introduce the elephant random walk (ERW) with memory consisting of randomly selected steps from its history. It is a time-changed variant of the standard elephant random walk with memory consisting of its full history. At each time point, the time changing component is the composition of two uniformly distributed independent random variables with support over all the past steps. Several conditional distributional properties including the conditional mean increments and conditional displacement of ERW with random memory are obtained. Using these conditional results, we derive the recursive and explicit expressions for the mean increments and mean displacement of the walk.

math.PR↗

arXiv2024

Iterated Generalized Counting Process and its Extensions

In this paper, we study the composition of two independent GCPs which we call the iterated generalized counting process (IGCP). Its distributional properties such as the transition probabilities, probability generating function, state probabilities and its corresponding Lévy measure are obtained. We study some integrals of the IGCP. Also, we study some of its extensions, for example, the compound IGCP, the multivariate IGCP and the $q$-iterated GCP. It is shown that the IGCP and the compound IGCP are identically distributed to a compound GCP which leads to their martingale characterizations. Later, a time-changed version of the IGCP is considered where the time is changed by an inverse stable subordinator. Using its covariance structure, we establish that the time-changed IGCP exhibits long-range dependence property. Moreover, we show that its increment process exhibits short-range dependence property. Also, it is shown that its one-dimensional distributions are not infinitely divisible. Initially, some of its potential real life applications are discussed.

math.PR↗

arXiv2024

On the Multivariate Generalized Counting Process and its Time-Changed Variants

In this paper, we study a multivariate version of the generalized counting process (GCP) and discuss its various time-changed variants. The time is changed using random processes such as the stable subordinator, inverse stable subordinator, and their composition, tempered stable subordinator, gamma subordinator $etc.$ Several distributional properties that include the probability generating function, probability mass function and their governing differential equations are obtained for these variants. It is shown that some of these time-changed processes are Lévy and for such processes we have derived the associated Lévy measure. The explicit expressions for the covariance and codifference of the component processes for some of these time-changed variants are obtained. An application of the multivariate generalized space fractional counting process to shock models is discussed.

math.PR↗

arXiv2024

On the Superposition and Thinning of Generalized Counting Processes

In this paper, we study the merging and splitting of generalized counting processes (GCPs). First, we study the merging of a finite number of independent GCPs and then extend it to the case of countably infinite. The merged process is observed to be a GCP with increased arrival rates. It is shown that a packet of jumps arrives in the merged process according to the Poisson process. Also, we study two different types of splitting of a GCP. In the first type, we study the splitting of jumps of a GCP where the probability of simultaneous jumps in the split components is negligible. In the second type, we consider the splitting of jumps in which there is a possibility of simultaneous jumps in the split components. It is shown that the split components are GCPs with certain decreased jump rates. Moreover, the independence of split components is established. Later, we discuss applications of the obtained results to industrial fishing problem and hotel booking management system.

math.PR↗