Journal of Theoretical Probability

, Volume 29, Issue 4, pp 1485–1509

# Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

Article

## Abstract

Fix any $$n\ge 1$$. Let $$\tilde{X}_1,\ldots ,\tilde{X}_n$$ be independent random variables. For each $$1\le j \le n$$, $$\tilde{X}_j$$ is transformed in a canonical manner into a random variable $$X_j$$. The $$X_j$$ inherit independence from the $$\tilde{X}_j$$. Let $$s_y$$ and $$s_y^*$$ denote the upper $$\frac{1}{y}{\underline{\text{ th }}}$$ quantile of $$S_n=\sum _{j=1}^nX_j$$ and $$S^*_n=\sup _{1\le k\le n}S_k$$, respectively. We construct a computable quantity $$\underline{Q}_y$$ based on the marginal distributions of $$X_1,\ldots ,X_n$$ to produce upper and lower bounds for $$s_y$$ and $$s_y^*$$. We prove that for $$y\ge 8$$
\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}
where
\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}
and $$w_y$$ is the unique solution of
\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}
for $$w_y>\ln (\frac{y}{y-2})$$, and for $$y\ge 37$$
\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}
where
\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}
The distribution of $$S_n$$ is approximately centered around zero in that $$P(S_n\ge 0) \ge \frac{1}{18}$$ and $$P(S_n\le 0)\ge \frac{1}{65}$$. The results extend to $$n=\infty$$ if and only if for some (hence all) $$a>0$$
\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}
(1)

## Keywords

Sum of independent random variables Tail distributions Tail probabilities Quantile approximation  Hoffmann–Jørgensen/Klass–Nowicki inequality

## Mathematics Subject Classification (2010)

60G50 60E15 62G32

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