Weak laws of large numbers for nonnegative independent random variables without finite means. Issue 21 (2nd November 2019)

• Record Type: Journal Article
• Authors: Chen, Pingyan; Sung, Soo Hak
• Is Part Of: Communications in statistics. Volume 48:Issue 21(2019)
• Page Start: 5401
• Page End: 5413
• Publication Date: 2019-11-02
• Subjects: Weak law of large numbers -- Infinite mean -- Slow variation; 60F05 -- 62E20; Mathematical statistics -- Periodicals; Mathematics; Statistics; 519.2
• DOI: 10.1080/03610926.2018.1513145
• Languages: English
• ISSNs: 0361-0926

Abstract

Abstract: For a sequence of nonnegative independent random variables { X n, n ≥ 1 } with P ( X n > x ) ≈ l n ( x ) / x, where l n ( x ) > 0, n ≥ 1, are slowly varying functions, we present some mild conditions on { l n ( x ) } under which ∑ i = 1 n X i / b n → 1 in probability, where { b n, n ≥ 1 } is a norming sequence of positive numbers with b n → ∞ When the distributions of the random variables are specified more precisely, the norming sequence can be obtained more concretely. As special cases, we obtain the results of Adler (2012, 2017 ) and Nakata (2016).