arXiv subjects

Palaniappan Vellaisamy

Publications and source records attributed to Palaniappan Vellaisamy.

13 recordsLinked to original sources

arXiv2026

A Unified Approach to Beta Moments, Combinatorial Identities, and Random Walks

The study of random walks has increasingly been popular across diverse disciplines such as statistics, mathematics, quantum physics, where they are used to model paths consisting of successive random steps in a mathematical space. A fundamental quantity of interest is the probability that a simple symmetric random walk returns to the origin after 2n steps. In this paper, we develop a unified probabilistic approach that connects the return probabilities in arbitrary dimensions with moment representations. Using this framework, we provide probabilistic proofs of several combinatorial identities involving beta and gamma functions, and derive new combinatorial identities in general dimensions.

math.PR↗

arXiv2026

Generalization and Probabilistic Proofs of Some Combinatorial Identities

Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special functions. In particular, by studying moments of the difference of two gamma and beta random variables, both in the dependent and independent cases, we obtain new combinatorial identities. This approach provides a systematic method to derive further combinatorial identities from probabilistic transformations.

math.PR↗

arXiv2026

Linear combination of bilateral gamma random variables: distributional theory and approximations

In this article, we obtain the exact distribution of a linear combination of bilateral gamma (BG) random variables (r.v.s). Next, we discuss the distributional properties of the linear combination of BG r.v.s, including probability density function, cumulant generating function and characteristic function. A Stein characterization is developed, which leads us to several distributional approximation results with explicit error bounds in both Kolmogorov and Wasserstein distances. Related limit theorems are also discussed. Furthermore, we show that the associated Lévy processes are finite-variation processes with BG distributed increments having random parameters. Finally, we apply our results in exponential stock models.

math.PR↗

arXiv2025

Time-changed generalized fractional Skellam process

In this paper, we introduce and study two time-changed variants of the generalized fractional Skellam process. These are obtained by time-changing the generalized fractional Skellam process with an independent Lévy subordinator with finite moments of any order and its inverse, respectively. We call the introduced processes the time-changed generalized fractional Skellam process-I (TCGFSP-I) and the time-changed generalized fractional Skellam process-II (TCGFSP-II), respectively. The probability generating function, moment generating function, moments, factorial moments, variance, covariance, {\it etc.}, are derived for the TCGFSP-I. We obtain a variant of the law of the iterated logarithm for it and establish its long-range dependence property. Several special cases of the TCGFSP-I are considered, and the associated system of governing differential equations is obtained. Later, some distributional properties and particular cases are discussed for the TCGFSP-II.

math.PR↗

arXiv2025

A New Compound Poisson Process and Its Fractional Versions

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 representation for its distribution is obtained in terms of Bell polynomials. We then extend it to a compound Poisson process and time fractional compound Poisson process (TFCPP). It is shown that the one-dimensional distributions of the TFCPP exhibit over-dispersion property, are not infinitely divisible and possess the long-range dependence property. Also, their moments and factorial moments are derived. Finally, the fractional differential equation associated with the TFCPP is also obtained.

math.PR↗

arXiv2024

Bilateral Gamma Approximation in Weiner Space

This paper deals with bilateral-gamma (BG) approximation to functionals of an isonormal Gaussian process. We use Malliavin-Stein method to obtain the error bounds for the smooth Wasserstein distance. As by-products, the error bounds for variance-gamma (V G), Laplace, gamma and normal approximations are presented. Our approach is new in the sense that the Stein equation is based on integral operators rather than diferential operators commonly used in the literature. Some of our bounds are sharper than the existing ones. For the approximation of a random element from the second Wiener chaos to a BG distribution, the bounds are obtained in terms of their cumulants. Using this result, we show that a sequence of random variables (rvs) in the second Wiener chaos converges in distribution to a BG rv if their cumulants of order two to six converge. As an application of our results, we consider an approximation of homogeneous sums of independent rvs to a BG distribution, and mention some related limit theorems also. Finally, an approximation of a U-statistic to the BG distribution is discussed.

math.PR↗

arXiv2024

Approximations Related to Tempered Stable Distributions

In this article, we first obtain, for the Kolmogorov distance, an error bound between a tempered stable and a compound Poisson distribution and also an error bound between a tempered stable and an alpha stable distribution via Stein method. For the smooth Wasserstein distance, an error bound between two tempered stable distributions is also derived. As examples, we discuss the approximation of a tempered stable to normal and variance gamma distributions. As corollaries, the corresponding limit theorem follows

math.PR↗

arXiv2024

Covariance Identities and Variance Bounds for Infinitely Divisible Random Variables and Their Applications

In this article, we establish a general covariance identity for infinitely divisible distributions (IDD). Using this result, we derive Cacoullos type variance bounds for the IDD. Applications to some important distributions are discussed, in addition to the computation of variance bounds for certain posterior distributions. As another application, we derive the Stein-type identity for the IDD, which involves the L'evy measure. This result in turn is used to derive the Stein-type identity for the CGMY distributions and the variance-gamma distributions (VGD). This approach, especially for the VGD is new and simpler, compared to the ones available in the literature. Finally, as another nontrivial application, we apply the covariance identity in deriving known and some new formulas for the weighted premium calculation principles (WPCP) and Gini coefficient for the IDD.

math.PR↗

arXiv2019

On large deviations for sums of discrete m-dependent random variables

The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$ is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a statistic of 2-runs and $Z_n$ is negative binomial rv; and (c) when $S_n$ is statistic of $N(1,1)$-events and $Z_n$ is a binomial r.v. We also consider the approximation of $P(S_n\geqslant x)$ by Poisson distribution with parameter depending on $x$.

math.ST↗

arXiv2018

Analysis of intersections of trajectories of linear systems

Present article deals with trajectorial intersections in linear fractional systems ('systems'). We propose a classification of intersections of trajectories in three classes viz. trajectories intersecting at same time(EIST), trajectories intersecting at distinct times(EIDT) and self intersections of a trajectory. We prove a generalization of separation theorem for the case of linear fractional systems. This result proves existence of EIST. Based on the presence of EIST, systems are further classified in two types; Type I and Type II systems, which are analyzed further for EIDT. Besides constant solutions and limit-cycle behavior, a fractional trajectory can have nodal or cuspoidal intersections with itself. We give a necessary and sufficient condition for a trajectory to have such types of intersections.

math.DS↗

arXiv2018

On closeness of two discrete weighted sums

The effect that weighted summands have on each other in approximations of $S=w_1S_1+w_2S_2+\cdots+w_NS_N$ is investigated. Here, $S_i$'s are sums of integer-valued random variables, and $w_i$ denote weights, $i=1,\dots,N$. Two cases are considered: the general case of independent random variables when their closeness is ensured by the matching of factorial moments and the case when the $S_i$ has the Markov Binomial distribution. The Kolmogorov metric is used to estimate the accuracy of approximation.

math.PR↗

arXiv2011

Transient anomalous sub-diffusion on bounded domains

This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables, and eigenfunction expansions in time and space, are used to write strong solutions. Finally, stochastic solutions are written in terms of an inverse subordinator.

math.PR↗

arXiv2009

Distributed-order fractional Cauchy problems on bounded domains

In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. The fractional derivative models time delays in a diffusion process. The order of the fractional derivative can be distributed over the unit interval, to model a mixture of delay sources. In this paper, we provide explicit strong solutions and stochastic analogues for distributed-order fractional Cauchy problems on bounded domains with Dirichlet boundary conditions. Stochastic solutions are constructed using a non-Markovian time change of a killed Markov process generated by a uniformly elliptic second order space derivative operator.

math.PR↗