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Densities with Gaussian Tails
Author(s) -
Balkema A. A.,
Klüppelberg C.,
Resnick S. I.
Publication year - 1993
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-66.3.568
Subject(s) - mathematics , convolution (computer science) , convexity , gaussian , exponential function , constant (computer programming) , pure mathematics , exponential family , mathematical analysis , embedding , regular polygon , combinatorics , geometry , quantum mechanics , physics , machine learning , artificial neural network , computer science , financial economics , economics , programming language , artificial intelligence
Consider densities f i ( t ), for i = 1, …, d , on the real line which have thin tails in the sense that, for each i , f i ( t ) ∼ γ i ( t ) e −ψ i ( t ) , where γ i behaves roughly like a constant and ψ i is convex, C 2 , with ψ′ → ∞ and ψ″ > 0 and l/√ψ″ is self‐neglecting. (The latter is an asymptotic variation condition.) Then the convolution is of the same form f t * … * f d ( t ) ∼ γ( t ) e − ψ( t ) Formulae for γ, ψ are given in terms of the factor densities and involve the conjugate transform and infimal convolution of convexity theory. The derivations require embedding densities in exponential families and showing that the assumed form of the densities implies asymptotic normality of the exponential families.