Extremes of randomly scaled Gumbel risks

We investigate the product Y1Y2 of two independent positive risks Y1 and Y2. If Y1 has distribution in the Gumbel max-domain of attraction with some auxiliary function which is regularly varying at infinity and Y2 is bounded, then we show that Y1Y2 has also distribution in the Gumbel max-domain of attraction. If both Y1,Y2 have log-Weibullian or Weibullian tail behaviour, we prove that Y1Y2 has log-Weibullian or Weibullian asymptotic tail behaviour, respectively. We present here three theoretical applications concerned with a) the limit of point-wise maxima of randomly scaled Gaussian processes, b) extremes of Gaussian processes over random intervals, and c) the tail of supremum of iterated processes.