Asymptotics for Weighted Random Sums

By Mariana Olvera-Cravioto. Advances in Applied Probability, 2011

72 citations recorded by the source corpus.

Let {X i} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C 1, C 2,…) be a nonnegative random vector independent of the {X i } with N∈ℕ∪ {∞}. We study the weighted random sum S N =∑{i=1} N C i X i , and its maximum, M N =sup{1≤k N+1∑ i=1 k C i X i . This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(M N > x)∼ P(S N > x)∼ E[∑ i=1 N F̄(x/C i )] as x→∞. When E[X 1]>0 and the distribution of Z N =∑ i=1 N C i is also intermediate regularly varying, we obtain the asymptotics P(M N > x)∼ P(S N > x)∼ E[∑ i=1 N F̄}(x/C i )] +P(Z N > x/E[X 1]). For completeness, when the distribution of Z N is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(M N > x) ∼ P(S N > x)∼ P(Z N > x / E[X 1 ] hold.

Subfields: Artificial Intelligence, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7e0db104b4ec398b43f7aea5316f1b9d680b0d6e

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Citations in the corpus

Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions (2014, methodology): “We will also make use of a recent limit theorem in law for weighted sums proved in [3].”
Randomly weighted sums of subexponential random variables with application to capital allocation (2014, background): “See Shen et al. (2009), Zhang et al. (2009), Gao and Wang (2010), Chen et al. (2011), Yi et al. (2011), Fougères and Mercadier (2012), Hazra and Maulik (2012) and Olvera-Cravioto (2012), to name a few very recent ones.”