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- W3016488163 abstract "We consider the distributional fixed-point equation: $$R stackrel{mathcal{D}}{=} Q vee left( bigvee_{i=1}^N C_i R_i right),$$ where the ${R_i}$ are i.i.d.~copies of $R$, independent of the vector $(Q, N, {C_i})$, where $N in mathbb{N}$, $Q, {C_i} geq 0$ and $P(Q > 0) > 0$. By setting $W = log R$, $X_i = log C_i$, $Y = log Q$ it is equivalent to the high-order Lindley equation $$W stackrel{mathcal{D}}{=} maxleft{ Y, , max_{1 leq i leq N} (X_i + W_i) right}.$$ It is known that under Kesten assumptions, $$P(W > t) sim H e^{-alpha t}, qquad t to infty,$$ where $alpha>0$ solves the Cramer-Lundberg equation $E left[ sum_{j=1}^N C_i ^alpha right] = Eleft[ sum_{i=1}^N e^{alpha X_i} right] = 1$. The main goal of this paper is to provide an explicit representation of the constant $H$ which can be directly connected to the underlying weighted branching process where $W$ is constructed. The main tool in our analysis is Alsmeyer's Markov renewal theorem, which we apply after a suitable change of measure. Furthermore, our new explicit approach for deriving $H$ will also provide an unbiased and strongly efficient importance sampling estimator for the rare event probability $P(W > t)$. We provide numerical examples illustrating the use of this new algorithm." @default.
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- W3016488163 date "2020-04-19" @default.
- W3016488163 modified "2023-10-18" @default.
- W3016488163 title "On the renewal theorem for maxima on trees" @default.
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