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Limit distributions for the ratio of the random sum of squares to the square of the random sum with applications to risk measures. (English) Zbl 1164.60328

Let \(\{X_n,\,n\geq 1\}\) be a sequence of iid positive random variables having regularly varying tails with index \(-\alpha<0\), let \(\{N(t);\,t\geq 0\}\) be a counting process independent of the \(X_i\)’s and, for any fixed \(t\geq 0\), define \[ T_{N(t)}:=\frac{X_1^2+X_2^2+\cdots+X_{N(t)}^2}{(X_1+X_2+\cdots+X_{N(t)})^2} \] if \(N(t)\geq 1\) and \(T_{N(t)}:=0\) otherwise. Under the appropriate conditions on the counting process, the authors derive weak limits for \(T_{N(t)}\). These results are used in the second part of this paper to obtain the asymptotic behavior of two risk measures: the sample coefficient of variation and the sample dispersion.

MSC:

60F05 Central limit and other weak theorems
91B30 Risk theory, insurance (MSC2010)
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