Inversion for Heavy-tailedness of the Product of Two
Independent Random Variables
Yiqing Chen
a
a,∗
and Kam C. Yuen
b
Department of Mathematical Sciences,
The University of Liverpool, Liverpool, L69 7ZL, UK
b
Department of Statistics and Actuarial Science,
The University of Hong Kong, Pokfulam Road, Hong Kong
Abstract
Let X and Y be two independent and positive random variables and let Z be their
product. Denote by F , G and H the distribution functions of X, Y and Z, respectively.
This product is a basic structure in stochastic modeling in many applied areas. The
well-known Breiman’s theorem shows that if F is regularly varying with index α > 0
and EY β < ∞ for some β > α, then H(x) ∼ EY α F (x). This indicates that the
product Z inherits the tail behavior of X and the contribution of Y is reflected in the
coefficient EY α . Theorem 4.2 of Jacobsen et al. (1999) gives an inverse of Breiman’s
theorem. If H is regularly varying with index α > 0 and EY β < ∞ for some β > α,
then F ∈ R−α if (and only if) EY α+iθ 6= 0 for all θ ∈ R. We shall extend the study of
the inverse problem to some important larger classes of heavy-tailed distributions.
Keywords: asymptotics; Breiman’s theorem; heavy tails; tail probability; product
of independent random variables
References
[1] Breiman, L. On some limit theorems similar to the arc-sin law. Theory of Probability
and Its Applications 10 (1965), 323–331.
[2] Cline, D. B. H.; Samorodnitsky, G. Subexponentiality of the product of independent
random variables. Stochastic Processes and their Applications 49 (1994), no. 1, 75–98.
[3] Jacobsen, M.; Mikosch, T.; Rosiński, J.; Samorodnitsky, G. Inverse problems for regular
variation of linear filters, a cancellation property for σ-finite measures and identification
of stable laws. Annals of Applied Probability 19 (2009), no. 1, 210–242.
[4] Schmid, M.; Schneeweiss, H. The effect of microaggregation by individual ranking on
the estimation of moments. Journal of Econometrics 153 (2009), no. 2, 174–182.
[5] Tang, Q. From light tails to heavy tails through multiplier. Extremes 11 (2008), no. 4,
379–391.
∗
Speaker: Yiqing Chen; E-mail: [email protected]; Tel.: 44-151-794-4749; Fax: 44-151-794-4061
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