Acta Mathematica Academiae Scientiarum Hungaricae 40. (1982)

1982 / 1-2. szám - Longnecker, M. T.: Generalized SLLN under weak multiplicative dependence restrictions

Acta Math. Acad. Sei. Hurtgar. 40 (1—2), (1982), 153—157. GENERALIZED SLLN UNDER WEAK MULTIPLICATIVE DEPENDENCE RESTRICTIONS By M. T. LONGNECKER (College Station) 1. Introduction. For random variables Xlt X2, ... and constants аг, a2, put П Sn= 2 ak^k- The main result of this paper is a strong law of large numbers for {S',,} 1 with respect to a sequence of constants {b„}, where the dependence restrictions on the sequence {Xj} will be of the weak multiplicative type. A rate of convergence will be provided for the strong law. These results will broaden the theorems found in Móricz [3]. 2. Dependence restrictions and previous results. The term “weak multiplicative” refers to any form of restriction on the product moments E{XjlXj2...Xjv} of order v. Three different but related conditions were formulated in Longnecker and Serfling [2]. They will be stated in this section for completeness. The first two conditions are orthogonality related dependence restrictions. Definition. A sequence of random variables {X;} satisfies Condition A with respect to an even integer v, a sequence of constants {a}}, and a symmetric function g of v —1 arguments if (2.1a) IE {XhXjt...Xjv}\ =s g(h-ji,j3-b, -,L-h--dahah...ajv for all láftc.and if Definition. A sequence of random variables {Xj} satisfies Condition В with respect to an even integer v, a sequence of constants {a,}, and a symmetric function g of v/2 arguments if (2.2a) \Е{Х^...Х^}\ S g(j*2 —ji, ji —ja> ••■>./»~jy-i)aji• • • ayv for all 1 Sy1<...<yv, and if (2.2b) 2 2— 2 8Ü1, •••.Á/2-i, k) <=». k=lj\=l У»/,-!“! A third dependence restriction which is related to Gaussian time series is Definition. A sequence of random variables {Xj} satisfies Condition C with respect to an even integer v, constants {а7}, a function f(j), and a function g of (2.1b) 2 2 ••• 2 gO’i. -Jv-a.fc) <°°-4=1 A-1 Á.-1-1 Acta Mathematica Academiae Scientlarum Hungaricae iO, 1982

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