On the behavior of the product of independent random variables. (English) Zbl 1106.60018

Su, Chun; Chen, Yu

Sci. China, Ser. A 49, No. 3, 342-359 (2006).

Summary: For two independent nonnegative random variables \(X\) and \(Y\), we treat \(X\) as the initial variable of major importance and \(Y\) as a modifier (such as the interest rate of a portfolio). Stability in the tail behaviors of the product compared with that of the original variable \(X\) is of practical interest. We study the tail behaviors of the product \(XY\) when the distribution of \(X\) belongs to the classes \(\mathcal L\) and \(\mathcal S\), respectively. Under appropriate conditions, we show that the distribution of the product \(XY\) is in the same class as \(X\) when \(X\) belongs to class \(\mathcal L\) or \(\mathcal S\), in other words, classes \(\mathcal L\) and \(\mathcal S\) are stable under some mild conditions on the distribution of \(Y\). We also show that if the distribution of \(X\) is in class \(\mathcal L (\gamma)\) \((\gamma >0)\) and continuous, then the product \(XY\) is in \(\mathcal L\) if and only if \(Y\) is unbounded.

MSC:

60E05

Probability distributions: general theory

Keywords:

independent product; stability; tails of a distribution; heavy-tailed

References:

[1]	Embrechts, P., Klüppelberg, C., Mikosch, T., Modelling Extremal Events for Insurance and Finance, Berlin: Springer, 1997. · Zbl 0873.62116
[2]	Tang, Q., Tsitsiashvili, G., Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Processes and Their Applications, 2003, 108: 299–325. · Zbl 1075.91563
[3]	Cline, D. B. H., Samorodnitsky, G., Subexponentiality of the product of independent random variables, Stochastic Process and Their Applications, 1994, 49: 75–98. · Zbl 0799.60015
[4]	Embrechts, P., Goldie, C. M., On closure and factorization properties of subexponential distribution, J. Austral. Math. Soc. Ser. A, 1980, 29: 243–256. · Zbl 0425.60011
[5]	Shimura, T., The product of independent random variables with slowly varying truncated moments, J. Austral. Math. Soc. Ser. A, 1997, 62: 186–197. · Zbl 0894.60013
[6]	Su, C., Chen, J., Hu, Z., Some discussions on the class L({\(\gamma\)}), Journal of Mathematical Science, 2004, 122(4): 3416–3425. · Zbl 1066.60017
[7]	Pitman, E. J. G., Subexponential distribution functions, J. Austral. Math. Soc. Ser. A, 1980, 29: 337–347. · Zbl 0425.60012
[8]	Su, C., Tang Q., Characterizations on Heavy-tailed Distributions by Means of Hazard Rate, Acta Mathematicae Applicatae Sinica, English Series, 2003, 19(1): 135–142. · Zbl 1043.60012
[9]	Su, C., Tang, Q., Heavy-tailed distributions and their application, in Probability, Finance and Insurance, Proceedings of a Workshop at the University of Hong Kong (eds. Lai, T. L., Yang, H. L., Yung, S. P.), Singapore: World Scientific, 2004, 218–236.