arXiv subjects

Kalyan Barman

Publications and source records attributed to Kalyan Barman.

6 recordsLinked to original sources

arXiv2026

Variance gamma approximation to sums of triplewise independent random variables

In this article, we first discuss how triplewise independent random variables (rvs) are connected to a complete bipartite graph. Using the connection, we construct a sequence of triplewise independent rvs. We next consider a variance gamma (VG) approximation of sums of such triplewise independent rvs. Using Stein's method and the generalized zero-bias transformation, we obtain our bounds. Related limit theorems are also discussed.

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↗

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

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↗

arXiv2022

A Unified Approach to Stein's Method for Stable Distributions

In this article, we first review the connection between Lévy processes and infinitely divisible random variables, and the classification of infinitely divisible distributions. Using this connection and the Lévy-Khinchine representation of the characteristic function, we establish a Stein identity for an infinitely divisible random variable. The classification and slight modification in approach give us a Stein identity for an $α$-stable random variable with $α\in (0,2).$ Using fine regularity estimates for the solution to Stein equation, we derive error bounds for $α$-stable approximations. We then apply these results to obtain rates of convergence. Finally, we compare these rates with the results available in the literature.

math.PR↗

arXiv2022

Stein's Method for Tempered Stable Distributions

In this article, we develop Stein characterization for two-sided tempered stable distribution. Stein characterizations for normal, gamma, Laplace, and variance-gamma distributions already known in the literature follow easily. One can also derive Stein characterizations for more difficult distributions such as the distribution of product of two normal random variables, a difference between two gamma random variables. Using the semigroup approach, we obtain estimates of the solution to Stein equation. Finally, we apply these estimates to obtain error bounds in the Wasserstein-type distance for tempered stable approximation in three well-known problems: comparison between two tempered stable distributions, Laplace approximation of random geometric sums, and six moment theorem for the symmetric variance-gamma approximation of functionals of double Wiener-It$\ddot{\text{o}}$ integrals. We also compare our results with the existing literature.

math.PR↗