Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

Michael J. Klass () and Krzysztof Nowicki ()
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Michael J. Klass: University of California, Berkeley
Krzysztof Nowicki: Lund University

Journal of Theoretical Probability, 2016, vol. 29, issue 4, 1485-1509

Abstract: Abstract Fix any $$n\ge 1$$ n ≥ 1 . Let $$\tilde{X}_1,\ldots ,\tilde{X}_n$$ X ~ 1 , … , X ~ n be independent random variables. For each $$1\le j \le n$$ 1 ≤ j ≤ n , $$\tilde{X}_j$$ X ~ j is transformed in a canonical manner into a random variable $$X_j$$ X j . The $$X_j$$ X j inherit independence from the $$\tilde{X}_j$$ X ~ j . Let $$s_y$$ s y and $$s_y^*$$ s y ∗ denote the upper $$\frac{1}{y}{\underline{\text{ th }}}$$ 1 y th ̲ quantile of $$S_n=\sum _{j=1}^nX_j$$ S n = ∑ j = 1 n X j and $$S^*_n=\sup _{1\le k\le n}S_k$$ S n ∗ = sup 1 ≤ k ≤ n S k , respectively. We construct a computable quantity $$\underline{Q}_y$$ Q ̲ y based on the marginal distributions of $$X_1,\ldots ,X_n$$ X 1 , … , X n to produce upper and lower bounds for $$s_y$$ s y and $$s_y^*$$ s y ∗ . We prove that for $$y\ge 8$$ y ≥ 8 $$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$ 6 - 1 γ 3 y / 16 Q ̲ 3 y / 16 ≤ s y ∗ ≤ Q ̲ y where $$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$ γ y = 1 2 w y + 1 and $$w_y$$ w y is the unique solution of $$\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}$$ ( w y e ln ( y y - 2 ) ) w y = 2 y - 4 for $$w_y>\ln (\frac{y}{y-2})$$ w y > ln ( y y - 2 ) , and for $$y\ge 37$$ y ≥ 37 $$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)} 0$$ a > 0 1 $$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}

Keywords: Sum of independent random variables; Tail distributions; Tail probabilities; Quantile approximation; Hoffmann–Jørgensen/Klass–Nowicki inequality; 60G50; 60E15; 62G32 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-015-0615-y

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