Volume 12, Issue 3
Complete and Complete Moment Convergence of the Weighted Sums of $ρ^∗$-Mixing Random Vectors in Hilbert Spaces

East Asian J. Appl. Math., 12 (2022), pp. 617-627.

Published online: 2022-04

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• Abstract

Let $1 ≤ p < 2$, $\alpha > p$, $\{a_{ni}, 1 ≤ i ≤ n, n ≥ 1\}$ be a set of real numbers with the property ${\rm sup}_{n≥1} n^{−1} \sum _{i=1}^n |a_{ni}|^α < ∞$ and let $\{X, X_n , n ≥ 1\}$ 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 $\epsilon > 0$ the following inequalities hold: $$\sum\limits_{n=1}^∞ n^{-1}p \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|>\epsilon n^{\frac{1}{p}}\right)<∞,$$  $$\sum\limits_{n=1}^∞ n^{-1-\frac{1}{p}}E \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|-\epsilon n^{\frac{1}{p}}\right)^+<∞.$$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.

• Keywords

Complete convergence, $ρ^∗$-mixing random vectors, complete moment convergence, weighted sums, Marcinkiewicz-Zygmund type strong law of large numbers.

60F15

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@Article{EAJAM-12-617, author = {}, title = {Complete and Complete Moment Convergence of the Weighted Sums of $ρ^∗$-Mixing Random Vectors in Hilbert Spaces}, journal = {East Asian Journal on Applied Mathematics}, year = {2022}, volume = {12}, number = {3}, pages = {617--627}, abstract = {

Let $1 ≤ p < 2$, $\alpha > p$, $\{a_{ni}, 1 ≤ i ≤ n, n ≥ 1\}$ be a set of real numbers with the property ${\rm sup}_{n≥1} n^{−1} \sum _{i=1}^n |a_{ni}|^α < ∞$ and let $\{X, X_n , n ≥ 1\}$ 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 $\epsilon > 0$ the following inequalities hold: $$\sum\limits_{n=1}^∞ n^{-1}p \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|>\epsilon n^{\frac{1}{p}}\right)<∞,$$  $$\sum\limits_{n=1}^∞ n^{-1-\frac{1}{p}}E \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|-\epsilon n^{\frac{1}{p}}\right)^+<∞.$$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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.010821.131221}, url = {http://global-sci.org/intro/article_detail/eajam/20410.html} }
TY - JOUR T1 - Complete and Complete Moment Convergence of the Weighted Sums of $ρ^∗$-Mixing Random Vectors in Hilbert Spaces JO - East Asian Journal on Applied Mathematics VL - 3 SP - 617 EP - 627 PY - 2022 DA - 2022/04 SN - 12 DO - http://doi.org/10.4208/eajam.010821.131221 UR - https://global-sci.org/intro/article_detail/eajam/20410.html KW - Complete convergence, $ρ^∗$-mixing random vectors, complete moment convergence, weighted sums, Marcinkiewicz-Zygmund type strong law of large numbers. AB -

Let $1 ≤ p < 2$, $\alpha > p$, $\{a_{ni}, 1 ≤ i ≤ n, n ≥ 1\}$ be a set of real numbers with the property ${\rm sup}_{n≥1} n^{−1} \sum _{i=1}^n |a_{ni}|^α < ∞$ and let $\{X, X_n , n ≥ 1\}$ 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 $\epsilon > 0$ the following inequalities hold: $$\sum\limits_{n=1}^∞ n^{-1}p \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|>\epsilon n^{\frac{1}{p}}\right)<∞,$$  $$\sum\limits_{n=1}^∞ n^{-1-\frac{1}{p}}E \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|-\epsilon n^{\frac{1}{p}}\right)^+<∞.$$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.

Mi Hwa Ko. (2022). Complete and Complete Moment Convergence of the Weighted Sums of $ρ^∗$-Mixing Random Vectors in Hilbert Spaces. East Asian Journal on Applied Mathematics. 12 (3). 617-627. doi:10.4208/eajam.010821.131221
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