arXiv subjects

Manisha Dhillon

Publications and source records attributed to Manisha Dhillon.

9 recordsLinked to original sources

arXiv2026

Amnesic Elephant Random Walk with Polynomially Decaying Step Sizes

In this paper, we introduce an amnesic elephant random walk with polynomially decaying step sizes. The walk retains the loss of memory effect of amnesic ERW, however its step sizes decay polynomially over time. We study the effect of step-size exponent and memory parameter on the long-time behaviour of the walk. Two critical thresholds are identified that determine its phase diagram. We obtain almost sure convergence results, law of iterated logarithm, asymptotic normality and mean square displacement rate of the walk across different parameter regimes. The polynomial decay of step sizes gives rise to a subdiffusive regime while classical diffusion is recovered in the absence of decay. Also, we identify localization regimes in which the walk converges almost surely to a finite random variable.

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↗

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 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↗