Journal of Theoretical Probability

, Volume 29, Issue 4, pp 1485–1509 | Cite as

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

  • Michael J. Klass
  • Krzysztof Nowicki


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 \(y\ge 8\)
$$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$
$$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$
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 \(y\ge 37\)
$$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}$$
$$\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}$$
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) \(a>0\)
$$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}$$


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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Departments of Statistics and MathematicsUniversity of California, BerkeleyBerkeleyUSA
  2. 2.Department of StatisticsLund UniversityLundSweden

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