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June 2006 On convolution equivalence with applications
Qihe Tang
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Bernoulli 12(3): 535-549 (June 2006). DOI: 10.3150/bj/1151525135


A distribution F on ( -,) is said to belong to the class S (γ) for some γ 0 if lim x F ¯( x-u )/F ¯(x)=e γ u holds for all u and lim x F * 2 ¯(x)F ¯(x)=2m F exists and is finite. Let X and Y be two independent random variables, where X has a distribution in the class S (γ) and Y is non-negative with an endpoint y ̂ =sup{ y:P (Yy)<1 }(0,) . We prove that the product XY has a distribution in the class S (γ/y ̂) . We further apply this result to investigate the tail probabilities of Poisson shot noise processes and certain stochastic equations with random coefficients.


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Qihe Tang. "On convolution equivalence with applications." Bernoulli 12 (3) 535 - 549, June 2006.


Published: June 2006
First available in Project Euclid: 28 June 2006

zbMATH: 1114.60015
MathSciNet: MR2232731
Digital Object Identifier: 10.3150/bj/1151525135

Keywords: asymptotics , class \mathcal{S}(γ) , endpoint , Poisson shot noise , rapid variation , Stochastic equation , uniformity

Rights: Copyright © 2006 Bernoulli Society for Mathematical Statistics and Probability


Vol.12 • No. 3 • June 2006
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