Electronic Journal of Probability

Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case

Michael Klass and Krzysztof Nowicki

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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.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 47, 1276-1298.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818518

Digital Object Identifier
doi:10.1214/EJP.v12-452

Mathematical Reviews number (MathSciNet)
MR2346512

Zentralblatt MATH identifier
1127.60043

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60E15: Inequalities; stochastic orderings
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B09: Probabilistic methods in Banach space theory [See also 60Bxx]

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Klass, Michael; Nowicki, Krzysztof. Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case. Electron. J. Probab. 12 (2007), paper no. 47, 1276--1298. doi:10.1214/EJP.v12-452. https://projecteuclid.org/euclid.ejp/1464818518


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