Asymptotics for Weighted Random Sums
نویسنده
چکیده
Let {Xi} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄ . Let (N,C1, C2, . . .) be a nonnegative random vector independent of the {Xi} with N ∈ N ∪ {∞}. We study the weighted random sum SN = ∑Ni=1 CiXi , and its maximum, MN = sup1≤k x) ∼ P(SN > x) ∼ E[∑Ni=1 F̄ (x/Ci)] as x → ∞. When E[X1] > 0 and the distribution of ZN = ∑Ni=1 Ci is also intermediate regularly varying, we obtain the asymptotics P(MN > x) ∼ P(SN > x) ∼ E[∑Ni=1 F̄ (x/Ci)] + P(ZN > x/E[X1]). For completeness, when the distribution of ZN is intermediate regularly varying and heavier than F̄ , we also obtain conditions under which the asymptotic relations P(MN > x) ∼ P(SN > x) ∼ P(ZN > x/E[X1]) hold.
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