Abstract
Let \(\left\{ X,X_{i},i=1,2,...\right\} \) denote independent positive random variables having common distribution function (d.f.) F(x) and, independent of X, let ν denote an integer valued random variable. Using X 0=0, the random sum Z=∑ i=0 ν X i has d.f. \(G(x)=\sum_{n=0}^{\infty }\Pr\{\nu =n\}F^{n\ast }(x)\) where F n∗(x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each ε>0 we can find a constant K such that the inequality
holds. When F is subexponential and E(1 +ε)ν<∞, it is a standard result in risk theory that G(x) satisfies
In this paper, we show that (*) holds under weaker assumptions on ν and under stronger conditions on F. Stam (Adv. Appl. Prob. 5:308–327, 1973) considered the case where \( \overline{F}(x)=1-F(x)\) is regularly varying with index –α. He proved that if α>1 and \(E{\left( {\nu ^{{\alpha + \varepsilon }} } \right)} < \infty \), then relation (*) holds. For 0<α<1, it is sufficient that Eν<∞. In this paper we consider the case where \(\overline{F}(x)\) is an O-regularly varying subexponential function. If the lower Matuszewska index \(\beta (\overline{F})<-1\), then the condition \({\text{E}}{\left( {\nu ^{{{\left| {\beta {\left( {\overline{F} } \right)}} \right|} + 1 + \varepsilon }} } \right)} < \infty\) is sufficient for (*). If \(\beta (\overline{F} )>-1\), then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio \(\overline{F^{n\ast }}(x)/\overline{F} (x)\). In the paper, we also consider (*) in the special case where X is a positive stable random variable or has a compound Poisson distribution derived from such a random variable and, in this case, we show that for n≥2, the ratio \(\overline{F^{n\ast }}(x)/\overline{F}(x)\uparrow n\) as x↑∞. In Section 3 of the paper, we briefly discuss an extension of Kesten’s inequality. In the final section of the paper, we discuss a multivariate analogue of (*).
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Daley, D.J., Omey, E. & Vesilo, R. The tail behaviour of a random sum of subexponential random variables and vectors. Extremes 10, 21–39 (2007). https://doi.org/10.1007/s10687-007-0033-3
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DOI: https://doi.org/10.1007/s10687-007-0033-3
Keywords
- Heavy tails
- Subexponential distribution
- Regularly varying distribution
- O-regularly varying distribution
- Stable distribution
- Bounds
- Monotonicity