On convolution equivalence with applications. (English) Zbl 1114.60015

Tang, Qihe

Bernoulli 12, No. 3, 535-549 (2006).

Summary: A distribution \(F\) on \((-\infty,\infty)\) is said to belong to the class \({\mathcal S}(\gamma)\) for some \(\gamma\geq 0\) if \(\lim_{x\to\infty} \overline{F}(x-u)/ \overline{F}(x)= e^{\gamma u}\) holds for all \(u\) and \(\lim_{x\to\infty} \overline{F^{*2}}(x)/ \overline{F}(x)= 2m_F\) exists and is finite. Let \(X\) and \(Y\) be two independent random variables, where \(X\) has a distribution in the class \({\mathcal S}(\gamma)\) and \(Y\) is nonnegative with an endpoint \(\widehat{y}= \sup \{y: P(Y\leq y)< 1\}\in (0,\infty)\). We prove that the product \(XY\) has a distribution in the class \({\mathcal S}(\gamma/\widehat{y})\). We further apply this result to investigate the tail probabilities of Poisson shot noise processes and certain stochastic equations with random coefficients.

MSC:

60E05	Probability distributions: general theory
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60B15	Probability measures on groups or semigroups, Fourier transforms, factorization

Keywords:

asymptotics; class \(S(\gamma)\); endpoint; Poisson shot noise; rapid variation; stochastic equation; uniformity

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