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2007 Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case
Michael Klass, Krzysztof Nowicki
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Electron. J. Probab. 12: 1276-1298 (2007). DOI: 10.1214/EJP.v12-452

Abstract

Let $X_1, X_2, \dots$ be independent and symmetric random variables such that $S_n = X_1 + \cdots + X_n$ converges to a finite valued random variable $S$ a.s. and let $S^* = \sup_{1 \leq n \leq \infty} S_n$ (which is finite a.s.). We construct upper and lower bounds for $s_y$ and $s_y^*$, the upper $1/y$-th quantile of $S_y$ and $S^*$, respectively. Our approximations rely on an explicitly computable quantity $\underline q_y$ for which we prove that $$\frac 1 2 \underline q_{y/2} < s_y^* < 2 \underline q_{2y} \quad \text{ and } \quad \frac 1 2 \underline q_{ (y/4) ( 1 + \sqrt{ 1 - 8/y})} < s_y < 2 \underline q_{2y}. $$ The RHS's hold for $y \geq 2$ and the LHS's for $y \geq 94$ and $y \geq 97$, respectively. Although our results are derived primarily for symmetric random variables, they apply to non-negative variates and extend to an absolute value of a sum of independent but otherwise arbitrary random variables.

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Michael Klass. Krzysztof Nowicki. "Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case." Electron. J. Probab. 12 1276 - 1298, 2007. https://doi.org/10.1214/EJP.v12-452

Information

Accepted: 16 October 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1127.60043
MathSciNet: MR2346512
Digital Object Identifier: 10.1214/EJP.v12-452

Subjects:
Primary: 60E15 , 60G50
Secondary: 46B09 , 46E30

Keywords: Hoffmann-Jo rgensen/Klass-Nowicki Inequality , quantile approximation , Sum of independent rv's , tail distributions , tail distributions , tail probabilities

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