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
SN - 0894-9840
VL - 17
SP - 165
EP - 181
JO - Journal of Theoretical Probability
JF - Journal of Theoretical Probability
IS - 1
ER -