On the concentration of measure phenomenon for stable and related random vectors
نویسندگان
چکیده
where m(f(X)) is a median of f(X) and where Φ is the (one–dimensional) standard normal distribution function. The inequality (1) has seen many extensions and to date, most of the conditions under which these developments hold require the existence of finite exponential moments for the underlying vector X . It is thus natural to explore the robustness of this “concentration phenomenon” and to study the corresponding results for stable vectors. It is the purpose of these notes to initiate this study and to present a few concentration results for stable and related vectors, freeing us from the exponential moment requirement. Our main result will imply that if X is an α–stable random vector in R, then for all x > 0, P (f(X)−m(f(X)) ≥ x) ≤ 1 ∧ C(α, d) xα , (2)
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