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 - <p style="text-align: justify;">Let $1 ≤ p &lt; 2$, $\alpha &gt; p$, $\{a_{ni}, 1 ≤ i ≤ n, n ≥ 1\}$ be a set of real numbers with the
property ${\rm sup}_{n≥1} n^{−1} \sum _{i=1}^n |a_{ni}|^α &lt; ∞$ 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 $\epsilon &gt; 0$ the following inequalities hold: $$\sum\limits_{n=1}^∞ n^{-1}p \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|&gt;\epsilon n^{\frac{1}{p}}\right)&lt;∞,$$&nbsp; <span style="text-align: justify;">$$\sum\limits_{n=1}^∞ n^{-1-\frac{1}{p}}E \left(\max\limits_{1 \leq k \leq n}\left\|\sum\limits_{i=1}^k a_{ni}X_{i}\right\|-\epsilon n^{\frac{1}{p}}\right)^+&lt;∞.$$</span>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.</p>