Asymptotic results on the moments of the ratio of the random sum of squares to the square of the random sum

Details Asymptotic results on the moments of the ratio of the random sum of squares to the square of the random sum Asymptotic results on the moments of the ratio of the random sum of squares to the square of the random sum

2005-05-12

arxiv.org/abs/math/0505265v1

Mathematics

Probability

14 pages

Scientific paper

Let \{X_1, X_2, ...\} be a sequence of positive independent and identically distributed random variables of Pareto-type with index \alpha>0 and let \{N(t); t\geq 0\} be a mixed Poisson process independent of the X_i's. For t\geq 0, define T_{N(t)}:=\frac{X_1^2 + X_2^2 + ... + X_{N(t)}^2} {(X_1 + X_2 + ... + X_{N(t)})^2} if N(t)\geq 1 and T_{N(t)}:=0 otherwise. We derive the limiting behavior of the k-th moment of T_{N(t)}, k\in\mathbb{N}, by using the theory of functions of regular variation and an integral representation for \mathbb{E}\{T_{N(t)}^k\}. We also point out the connection between T_{N(t)} and the sample coefficient of variation which is a popular risk measure in practical applications.

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