Some finitely additive versions of the strong law of large numbers

LetX be a non-empty set,H= X{su\t8, \gs = \lj{in1}x\lj{in2}x,σ=γ ₁×γ ₂×… be an independent strategy onH, and {Y _n} be a sequence of coordinate mappings onH. The following strong law in a finitely additive setting is proved: For some constantr≧1, if \GS _n=1 ^\t8 {\GS(\vbY _n \vb^2r )n ¹⁺ⁿ < \t8 andσ(Y _n)=0 for alln=1, 2, …, then \1n\gS{inj-1}/{sun} Y{inj}Y _jconverges to 0 withσ-measure 1 asn → ∞. The theorem is a generalization of Chung’s strong law in a coordinate representation process. Finally, Kolmogorov’s strong law in a finitely additive setting is proved by an application of the theorem. This research was based in part on the author’s doctoral dissertation submitted to the University of Minnesota, and was written with the partial support of the United States Army Grant DA-ARO-D-31-124-70-G-102.