Now showing 1 - 3 of 3
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    A unified approach to Stein’s method for stable distributions
    (01-01-2022) ;
    Barman, Kalyan
    In this article, we first review the connection between Lévy processes and infinitely divisible random variables, and discuss the classification of infinitely divisible distributions. Next, we establish a Stein identity for an infinitely divisible random variable via the Lévy-Khintchine representation of the characteristic function. In particular, we establish and unify the Stein identities for an α-stable random variable available in the existing literature. Next, we derive the solutions of the Stein equations and its regularity estimates. Further, we derive error bounds for α-stable approximations, and also obtain rates of convergence results in Wasserstein-δ, δ < α for α ∈ (0, 1) and Wasserstein-type distances for α ∈ (1, 2). Finally, we compare these results with existing literature.
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    Approximations related to the sums of m-dependent random variables
    (01-06-2022)
    Kumar, Amit N.
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    Vellaisamy, P.
    In this paper, we mainly focus on the sums of non-negative integer-valued 1-dependent random variables and its approximation to the power series distribution. We first discuss some relevant results for power series distribution such as the Stein operator, uniform and non-uniform bounds on the solution of the Stein equation. Using Stein’s method, we obtain error bounds for the approximation problem considered. The obtained results can also be applied to the sums of m-dependent random variables via appropriate rearrangements of random variables. As special cases, we discuss two applications, namely, 2-runs and (k1,k2)-runs, and compare our bounds with existing bounds.
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    Publication
    On discrete Gibbs measure approximation to runs
    (01-01-2022)
    Kumar, A. N.
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    A Stein operator for the runs is derived as a perturbation of an operator for discrete Gibbs measure. Due to this fact, using perturbation technique, the approximation results for runs arising from identical and non-identical Bernoulli trials are derived via Stein’s method. The bounds obtained are new and their importance is demonstrated through an interesting application.