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.