Abstract
Following a strategy recently developed by Ivan Nourdin and Giovanni Peccati, we provide a general technique to compare the tail of a given random variable to that of a reference distribution, and apply it to all reference distributions in the so-called Pearson class. This enables us to give concrete conditions to ensure upper and/or lower bounds on the random variable’s tail of various power or exponential types. The Nourdin-Peccati strategy analyzes the relation between Stein’s method and the Malliavin calculus, and is adapted to dealing with comparisons to the Gaussian law. By studying the behavior of the solution to general Stein equations in detail, we show that the strategy can be extended to comparisons to a wide class of laws, including all Pearson distributions.
To the memory of Prof. Paul Malliavin
Mathematics Subject Classification (2010). Primary 60H07; Secondary 60G15, 60E15.
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Eden, R., Viens, F. (2013). General Upper and Lower Tail Estimates Using Malliavin Calculus and Stein’s Equations. In: Dalang, R., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications VII. Progress in Probability, vol 67. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0545-2_3
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DOI: https://doi.org/10.1007/978-3-0348-0545-2_3
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