Complete and Complete Moment Convergence of the Weighted Sums of ρ∗-Mixing Random Vectors in Hilbert Spaces
Year: 2022
East Asian Journal on Applied Mathematics, Vol. 12 (2022), Iss. 3 : pp. 617–627
Abstract
Let 1≤p<2, α>p, {ani,1≤i≤n,n≥1} be a set of real numbers with the property supn≥1n−1∑ni=1|ani|α<∞ and let {X,Xn,n≥1} be a sequence of H-valued ρ∗-mixing random vectors coordinatewise stochastically upper dominated by a random vector X. We provide conditions such that for any ϵ>0 the following inequalities hold: ∞∑n=1n−1p(max1≤k≤n‖k∑i=1aniXi‖>ϵn1p)<∞, ∞∑n=1n−1−1pE(max1≤k≤n‖k∑i=1aniXi‖−ϵn1p)+<∞.These results generalize the results of Chen and Sung (cf. J. Ineq. Appl. 121, 1–16 (2018)) to the ρ∗-mixing random vectors in H. In addition, a Marcinkiewicz-Zygmund type strong law of ρ∗-mixing random vectors in H is presented.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/eajam.010821.131221
East Asian Journal on Applied Mathematics, Vol. 12 (2022), Iss. 3 : pp. 617–627
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 11
Keywords: Complete convergence ρ∗-mixing random vectors complete moment convergence weighted sums Marcinkiewicz-Zygmund type strong law of large numbers.