Abstract
We are interested in the tail behavior of the randomly weighted sum \( \sum _{i=1}^{n}\theta _{i}X_{i}\), in which the primary random variables X 1, …, X n are real valued, independent and subexponentially distributed, while the random weights 𝜃 1, …, 𝜃 n are nonnegative and arbitrarily dependent, but independent of X 1, …, X n . For various important cases, we prove that the tail probability of \(\sum _{i=1}^{n}\theta _{i}X_{i}\) is asymptotically equivalent to the sum of the tail probabilities of 𝜃 1 X 1, …, 𝜃 n X n , which complies with the principle of a single big jump. An application to capital allocation is proposed.
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Tang, Q., Yuan, Z. Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes 17, 467–493 (2014). https://doi.org/10.1007/s10687-014-0191-z
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DOI: https://doi.org/10.1007/s10687-014-0191-z