## Electronic Journal of Probability

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

#### 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

Rights

#### 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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