arXiv subjects

Kuldeep Kumar Kataria

Publications and source records attributed to Kuldeep Kumar Kataria.

12 recordsLinked to original sources

arXiv2026

On a Telegraph Process with Generalized Mittag-Leffler Waiting times and Velocity Driven by Random trials

We study a generalized telegraph process in which the velocity is governed by random trials by considering the specific distribution of waiting times. In this telegraph process, a particle moving on real line may change its direction whenever there is an arrival in a counting process. This direction change is driven by the outcomes of random trials. In the first case, random trials are independent and identically distributed, and waiting times have Mittag-Leffler distribution. In the second case, random trials follow P\'olya urn scheme and the first waiting time is generalized Mittag-Leffler distributed whereas other waiting times have Mittag-Leffler distribution. In both cases, we obtain the discrete component of their probability law. Also, absolutely continuous components of their conditional probability law given initial velocity are derived. The plots of absolutely continuous components of their probability law are compared for different parameters. Conditional on the initial velocity, the distributions of $n$th event time of counting processes associated with these telegraph processes are obtained.

math.PR↗

arXiv2026

On Elephant Random Walk with Delayed Amnesia

In this paper, we introduce a modified elephant random walk that exhibits a transition from a uniform memory mechanism to a selective amnesic memory mechanism. Using a vector martingale approach, we study the asymptotic behaviour of the walk across different parameter regimes. In the diffusive and critical regimes, we establish almost sure convergence results, laws of iterated logarithm, asymptotic normality of the walk, and the growth rate of mean square displacement. In the superdiffusive regime, we prove an almost sure convergence result and obtain the corresponding mean square displacement rate for the walk. Also, we study some almost sure convergence results for its center of mass. Later, we extend the model by incorporating random step sizes and obtain asymptotic results for it.

math.PR↗

arXiv2026

On multidimensional elephant random walk with stops and random step sizes

In this paper, we study the number of moves in a multidimensional elephant random walk with stops. We establish several convergence results for the number of moves, including the law of large numbers and the law of iterated logarithm. Using a martingale approach, we study the multidimensional elephant random walk with random step sizes. For this model, we obtain several almost sure convergence results for the number of moves, including the law of large numbers, the quadratic strong law, the law of iterated logarithm and the central limit theorem. Similar convergence results are derived for the multidimensional elephant random walk with random step sizes.

math.PR↗

arXiv2026

On Time-Changed Birth-Death Processes with Catastrophes

We study two time-changed variants of the birth-death process with catastrophe where the time-changing components are the first hitting times of the stable subordinator and the tempered stable subordinator. For both the processes, we derive the governing system of fractional differential equations for their state probabilities. The Laplace transforms of these state probabilities are obtained in terms of those of the corresponding time-changed birth-death processes without catastrophes. We obtain the distribution of catastrophe occurrence times as well as the sojourn times within non-zero states. We study distributional properties of the first visit time to state zero in a particular case. Also, the first occurrence time of an effective catastrophe is studied. Moreover, we study the time-changed linear birth-death processes with catastrophes, derive the explicit expressions for its state probabilities, expectation and variance. For a specific case, we compare the expectation plots across different parameter values and provide an algorithm for simulating sample paths with illustrative plots.

math.PR↗

arXiv2025

Multivariate Generalized Counting Process via Gamma Subordination

In this paper, we study a multivariate gamma subordinator whose components are independent gamma processes subject to a random time governed by an independent negative binomial process. We derive the explicit expressions for its joint Laplace-Stieltjes transform, its probability density function and the associated governing differential equations. Also, we study a time-changed variant of the multivariate generalized counting process where the time is changed by an independent multivariate gamma subordinator. For this time-changed process, we obtain the corresponding Lévy measure and probability mass function. Later, we discuss an application of the time-changed multivariate generalized counting process to a shock model.

math.PR↗

arXiv2025

On Multiparameter Generalized Counting Process and its Time-Changed Variants

We introduce and study a multiparameter version of the generalized counting process (GCP), where there is a possibility of finitely many arrivals simultaneously. We call it the multiparameter GCP. In a particular case, it is uniquely represented as a weighted sum of independent multiparameter Poisson processes. For a specific case, we establish a relationship between the multiparameter GCP and the sum of independent GCPs. Some of its time-changed variants are studied where the time-changing components used are the multiparameter stable subordinator and the multiparameter inverse stable subordinator. An integral of the multiparameter GCP is defined, and its asymptotic distribution is obtained.

math.PR↗

arXiv2025

Tempered Erlang Queue with Multiple Arrivals

In this paper, we introduce and study a time-changed variant of the Erlang queue with multiple arrivals where the time-changing component used is the first hitting time of a tempered stable subordinator. The system of fractional difference-differential equations that governs its state probabilities is derived which is solved to obtain their explicit expressions. An equivalent representation in terms of phases and the mean queue length is obtained. For a particular case, the distribution of inter-arrival times, inter-phase times, sojourn times, busy period and that of conditional waiting times are derived.

math.PR↗

arXiv2025

On Mixed Time-Changed Erlang Queue

We study a time-changed variant of the Erlang queue by taking the first hitting time of a mixed stable subordinator as the time-changing component. We call it the mixed time-changed Erlang queue. We derive the system of fractional differential equations that governs its state probabilities. The explicit expressions for the state probabilities of mixed time-changed Erlang queue and their Laplace transform are derived. Equivalently, it is represented in terms of phases and its mean queue length is obtained. Also, some distributional properties of the mixed time-changed Erlang queue such as the distribution of its inter-arrival times, inter-phase times, service times and busy period are derived. Later, its conditional waiting time is discussed and two plots of sample paths simulation are presented.

math.PR↗

arXiv2025

On Two Parameter Time-Changed Poisson Random Fields with Drifts

We study the composition of bivariate Lévy process with bivariate inverse subordinator. The explicit expressions for its dispersion and auto correlation matrices are obtained. Also, the time-changed two parameter Lévy processes with rectangular increments are studied. We introduce some time-changed variants of the Poisson random field in plane with and without drift, and derive the associated fractional differential equations for their distributions. Later, we consider some time-changed Lévy processes where the time-changing components are two parameter Poisson random fields with drifts. Moreover, two parameter coordinatewise semigroup operators associated with some of the introduced processes are discussed.

math.PR↗

arXiv2025

On a Fractional Variant of Linear Birth-Death Process

We introduce and study a fractional variant of the linear birth-death process, namely, the generalized fractional linear birth-death process (GFLBDP) which is defined by taking the regularized Hilfer-Prabhakar derivative in the system of differential equations that governs the state probabilities of linear birth-death process. For a particular choice of parameters, the GFLBDP reduces to the fractional linear birth-death process that involves the Caputo derivative. Its time-changed representation is obtained and utilized to derive the explicit expressions of its state probabilities. The explicit expressions for its mean and variance are derived. In a particular case, it is observed that the limiting distribution of the time changing process coincides to that of an inverse stable subordinator. A relation between the extinction probability of GFLBDP and the density of inter arrival times of a generalized fractional Poisson process is obtained. Later, we study some integrals of the GFLBDP and discuss the asymptotic distributional characteristics for a particular integral process. Also, an application of the path integral at random time to a genetic population with an upper bound is discussed.

math.PR↗

arXiv2025

Generalized Counting Process with Random Drift and Different Brownian Clocks

In this paper, we introduce drifted versions of the generalized counting process (GCP) with a deterministic drift and a random drift. The composition of stable subordinator with an independent inverse stable subordinator is taken as the random drift. We derive the probability law and its governing fractional differential equations for these drifted versions. Also, we study the GCP time-changed with different Brownian clocks, for example, the Brownian first passage-time with or without drift, elastic Brownian motion, Brownian sojourn time on positive half-line and the Bessel times. For these time-changed processes, we obtain the governing system of differential equation of their state probabilities, probability generating function, etc. Further, we consider a time-changed GCP where the time-change is done by subordinators linked to incomplete gamma function. Later, we study the fractional integral of GCP and its time-changed variant.

math.PR↗

arXiv2025

Time-changed multiparameter Poisson processes and martingales

We consider a multiparameter extension of the homogeneous Poisson counting process, namely, the multiparameter Poisson process (MPP). We derive its various distributional properties. Also, we consider an integral of the MPP and analyze its asymptotic distribution. Thereafter, we investigate three time-changed variants of the MPP, where the time-changing components are multivariate subordinator and inverse subordinators with both dependent and independent marginals. Later, we obtain some properties of the multiparameter martingales, which are then used to derive multiparameter martingale characterizations for the MPP and one of its time-changed variants.

math.PR↗