## 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.

### MSC:

 6e+06 Probability distributions: general theory
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### References:

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