Uniformly Accurate Quantile Bounds for Sums of Arbitrary Independent Random Variables

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

Journal of Theoretical Probability, 2010, vol. 23, issue 4, 1068-1091

Abstract: Abstract Fix any n≥1. Let X 1,…,X n be independent random variables such that S n =X 1+⋅⋅⋅+X n , and let $S^{*}_{n}=\sup_{1\le k\le n}S_{k}$ . We construct upper and lower bounds for s y and $s_{y}^{*}$ , the upper $\frac{1}{y}$ th quantiles of S n and $S^{*}_{n}$ , respectively. Our approximations rely on a computable quantity Q y and an explicit universal constant γ y , the latter depending only on y, for which we prove that $$\begin{array}{l}\displaystyle s_y\le s_y^*\le Q_y\quad\mbox{for }y>1,\\[4pt]\displaystyle \gamma_{3y/16}Q_{3y/16}-Q_1\le s_y^*\quad\mbox{for }y>\frac{32}{3},\end{array}$$ and $$\gamma_{u(y)}Q_{u(y)}-2Q_1\le s_y\quad \mbox{for }y>\frac{64}{3},$$ where $$u(y)=\frac{3y}{16}\biggl(\frac{1+\sqrt{1-\frac{64}{3y}}}{2}\biggr )\quad\mbox{and}\quad\gamma_y\to\frac{1}{3}\quad\mbox{as}\ y\to \infty.$$

Keywords: Sum of independent random variables; Tail distributions; Tail probabilities; Quantile approximation; Hoffmann-Jørgensen/Klass–Nowicki inequality; 60G50; 60E15; 46E30; 46B09 (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-009-0254-2

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