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

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• 作者：Michael J. Klass ; Krzysztof Nowicki
• 关键词：Sum of independent random variables ; Tail distributions ; Tail probabilities ; Quantile approximation ; Hoffmann–Jørgensen/Klass–Nowicki inequality
• 刊名：Journal of Theoretical Probability
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：29
• 期：4
• 页码：1485-1509
• 全文大小：496 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230
• 卷排序：29

文摘

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)KeywordsSum of independent random variablesTail distributionsTail probabilitiesQuantile approximation Hoffmann–Jørgensen/Klass–Nowicki inequalityMathematics Subject Classification (2010)60G5060E1562G32References1.Hahn, M.G., Klass, M.: Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential upper bounds. Ann. Probab. 25, 1451–1470 (1997)MathSciNetCrossRefMATHGoogle Scholar2.Hitczenko, P., Montgomery-Smith, S.: Measuring the magnitude of sums of independent random variables. Ann. Probab. 29, 447–466 (2001)MathSciNetCrossRefMATHGoogle Scholar3.Hofmann-Jørgensen, J.: Stochastic inequalities and perfect independence. J. Prog. Probab. 55, 3–34 (2003)MathSciNetMATHGoogle Scholar4.Jain, N.C., Pruitt, W.E.: Lower tail probability estimates for subordinates and decreasing random walks. Ann. Probab. 15, 75–102 (1987)MathSciNetCrossRefMATHGoogle Scholar5.Klass, M.: A method of approximating expectations of functions of sums of independent random variables. Ann. Probab. 9, 413–428 (1981)MathSciNetCrossRefMATHGoogle Scholar6.Klass, M., Nowicki, K.: Uniformly accurate quantile bounds via the truncated moment generating function: the symmetric case. Electron. J. Probab. 12, 1276–1298 (2007)MathSciNetCrossRefMATHGoogle Scholar7.Klass, M., Nowicki, K.: Uniformly accurate quantile bounds for sums of arbitrary independent random variables. J. Theor. Probab. 23, 1068–1091 (2010)MathSciNetCrossRefMATHGoogle Scholar8.Latała, R.: Estimation of moments of sums of independent real random variables. Ann. Probab. 25, 1502–1513 (1997)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer Science+Business Media New York 2015Authors and AffiliationsMichael J. Klass1Krzysztof Nowicki2Email author1.Departments of Statistics and MathematicsUniversity of California, BerkeleyBerkeleyUSA2.Department of StatisticsLund UniversityLundSweden About this article CrossMark Print ISSN 0894-9840 Online ISSN 1572-9230 Publisher Name Springer US About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips