Publication: A unified approach to Stein’s method for stable distributions
Date
01-01-2022
Authors
Neelesh Shankar Upadhye
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Abstract
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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Keywords
Semigroup approach., Stable approximation, Stable distributions, Stein’s method