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

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.


60E05 Probability distributions: general theory
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