Abstract
Let \(\{X_i, i\ge 1\}\) be i.i.d. \(\mathbb {R}^d\)-valued random vectors attracted to operator semi-stable laws and write \(S_n=\sum _{i=1}^{n}X_i\). This paper investigates precise large deviations for both the partial sums \(S_n\) and the random sums \(S_{N(t)}\), where N(t) is a counting process independent of the sequence \(\{X_i, i\ge 1\}\). In particular, we show for all unit vectors \(\theta \) the asymptotics
which holds uniformly for x-region \([\gamma _n, \infty )\), where \(\langle \cdot , \cdot \rangle \) is the standard inner product on \(\mathbb {R}^d\) and \(\{\gamma _n\}\) is some monotone sequence of positive numbers. As applications, the precise large deviations for random sums of real-valued random variables with regularly varying tails and \(\mathbb {R}^d\)-valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.
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The author wishes to express his deep gratitude to the referees for their valuable comments on an earlier version which improved the quality of this paper.
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This research was supported by NSFC Grant 11071076 and NSF of Zhejiang Province Grant LY14A010025.
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Wang, W. Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws. J Theor Probab 30, 64–84 (2017). https://doi.org/10.1007/s10959-015-0645-5
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DOI: https://doi.org/10.1007/s10959-015-0645-5
Keywords
- Operator semi-stable law
- Domain of attraction
- Precise large deviations
- Heavy tail
- Regular variation
- Random sums