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A LIL and limit distributions for trimmed sums of random vectors attracted to operator semi-stable laws

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Abstract

Let θ ∈ ℝd be a unit vector and let X,X 1,X 2, … be a sequence of i.i.d. ℝd-valued random vectors attracted to operator semi-stable laws. For each integer n ≥ 1, let X 1,n ≤ … ≤ X n,n denote the order statistics of X 1,X 2, …, X n according to priority of index, namely |〈X 1,n , θ〉| ≥ … ≥ |〈X n,n , θ〉|, where 〈·, ·〉 is an inner product on ℝd. For all integers r ≥ 0, define by (r) S n = Σ n−r i=1 X i,n the trimmed sum. In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums (r) S n . Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded. A stochastically compactness of (r) S n is obtained.

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Authors and Affiliations

  1. Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, P. R. China

    Wen Sheng Wang

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  1. Wen Sheng Wang
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Correspondence to Wen Sheng Wang.

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Supported by National Natural Science Foundation of China (Grant No. 11071076) and NSF of Zhejiang Province (Grant No. LY14A010025)

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Wang, W.S. A LIL and limit distributions for trimmed sums of random vectors attracted to operator semi-stable laws. Acta. Math. Sin.-English Ser. 30, 1555–1565 (2014). https://doi.org/10.1007/s10114-014-3487-7

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  • DOI: https://doi.org/10.1007/s10114-014-3487-7

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