TY - JOUR

T1 - SLLN for weighted independent identically distributed random variables

AU - Baxter, John

AU - Jones, Roger

AU - Lin, Michael

AU - Olsen, James

N1 - Funding Information:
We are grateful for the hospitality and support offered by North Dakota State University to the third author, and by the University of Minnesota and Ben-Gurion University to the last author. The last author’s travel to Israel was partially supported by an NDSU grant from the VP of Academic Affairs. Research of J. Olsen partially supported by ND-EPSCoR through NSF Grant EPS-9874802.

PY - 2004/1/1

Y1 - 2004/1/1

N2 - For any sequence {ak} with sup 1/n ∑k = 1 n |ak|q < ∞ for some q > 1, we prove that 1/n ∑k = 1n akXk converges to 0 a.s. for every {Xn} i.i.d. with E(|X1|) < ∞ and E(X1) = 0; the result is no longer true for q = 1, not even for the class of i.i.d. with X1 bounded. We also show that if {ak} is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {Xn} with finite absolute p th moment for some p > 1, 1/n ∑ k = 1n ak Xk converges a.s.

AB - For any sequence {ak} with sup 1/n ∑k = 1 n |ak|q < ∞ for some q > 1, we prove that 1/n ∑k = 1n akXk converges to 0 a.s. for every {Xn} i.i.d. with E(|X1|) < ∞ and E(X1) = 0; the result is no longer true for q = 1, not even for the class of i.i.d. with X1 bounded. We also show that if {ak} is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {Xn} with finite absolute p th moment for some p > 1, 1/n ∑ k = 1n ak Xk converges a.s.

KW - Besicovitch sequences

KW - Independent random variables

KW - Law of large numbers

KW - Weighted averages

UR - http://www.scopus.com/inward/record.url?scp=4043104179&partnerID=8YFLogxK

U2 - 10.1023/B:JOTP.0000020480.84425.8d

DO - 10.1023/B:JOTP.0000020480.84425.8d

M3 - Article

AN - SCOPUS:4043104179

VL - 17

SP - 165

EP - 181

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 1

ER -