@article{EJP452,
author = {Michael Klass and Krzysztof Nowicki},
title = {Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {12},
year = {2007},
keywords = {Sum of independent rv's, tail distributions, tail distributions,tail probabilities, quantile approximation, Hoffmann-Jo rgensen/Klass-Nowicki Inequality},
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.},
pages = {no. 47, 1276-1298},
issn = {1083-6489},
doi = {10.1214/EJP.v12-452},
url = {http://ejp.ejpecp.org/article/view/452}}