## Abstract

Fix any n≥1. Let X~1,…,X~n be independent random variables. For each 1≤j≤n, X~j is transformed in a canonical manner into a random variable Xj. The Xj inherit independence from the X~j. Let sy and s∗y denote the upper 1y th −−− quantile of Sn=∑nj=1Xj and S∗n=sup1≤k≤nSk, respectively. We construct a computable quantity Q−−y based on the marginal distributions of X1,…,Xn to produce upper and lower bounds for sy and s∗y. We prove that for y≥8

6−1γ3y/16Q−−3y/16≤s∗y≤Q−−y

where

γy=12wy+1

and wy is the unique solution of

(wyeln(yy−2))wy=2y−4

for wy>ln(yy−2), and for y≥37

19γu(y)Q−−u(y)<sy≤Q−−y

where

u(y)=3y32(1+1−643y−−−−−−√).

The distribution of Sn is approximately centered around zero in that P(Sn≥0)≥118 and P(Sn≤0)≥165. The results extend to n=∞ if and only if for some (hence all) a>0

∑j=1∞E{(X~j−mj)2∧a2}<∞.

6−1γ3y/16Q−−3y/16≤s∗y≤Q−−y

where

γy=12wy+1

and wy is the unique solution of

(wyeln(yy−2))wy=2y−4

for wy>ln(yy−2), and for y≥37

19γu(y)Q−−u(y)<sy≤Q−−y

where

u(y)=3y32(1+1−643y−−−−−−√).

The distribution of Sn is approximately centered around zero in that P(Sn≥0)≥118 and P(Sn≤0)≥165. The results extend to n=∞ if and only if for some (hence all) a>0

∑j=1∞E{(X~j−mj)2∧a2}<∞.

Original language | English |
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Pages (from-to) | 1-25 |

Journal | Journal of Theoretical Probability |

Volume | 28 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

## Subject classification (UKÄ)

- Probability Theory and Statistics

## Free keywords

- quantile approximation
- tail probabilities
- Sum of independent random variables
- tail distributions
- Hofmann-J/orgensen/Klass- Nowicki Inequality