Say that a triangle array $\{x_{n,i}:1\leq i\leq n, n\geq 1\}$ of random variables on some probability space $\{\Omega, \mathcal{F}, P\}$ is strong mixing if the coefficients $$\alpha_n(j)=\sup_{1\leq k\leq n-j}\sup\{\lvert P(A\cap B)-P(A)P(B)\rvert:A\in F_{1,n}^k, B\in F_{k+j,n}^n\}, \quad 0\leq j<n$$ satisfy $\alpha(j)=\sup_{n:0\leq j<n}\alpha_n(j)\to 0$ as $j\to\infty$ where $F_{i,n}^k= \sigma(x_{n,l}:i\leq l\leq k)$. Here we use the convention that $\alpha_{n}(j)=0$ for $j\geq n$ and $\alpha_{n}(j)=1/4$ for $j<0$.
My question: For each $n$, let $J_n\subseteq \{1,\dotsc,n\}$. Is $\{x_{n,i}1(i\in J_n):1\leq i\leq n, n\geq 1\}$ also strong mixing?
My attempt
Let $n=n_0$ be the smallest integer such that $J_n$ is nonempty. If $n<n_0$, Then $x_{n,i}=0$ for all $1\leq i \leq n$ (degenerate random variables, and thus, independent). In this case, $\alpha_n(j)=0, \forall j\geq 0$ for each $n< n_0$.
Let $\alpha'_T$ be the mixing coefficient of $\{x_{n,i}1(i\in J_n):1\leq i\leq n, n\geq 1\}$. If $n\geq n_0$, then \begin{align} F_{1,n}^k=\sigma (\cup_{l=1}^k\sigma(x_{n,l}))\supseteq \sigma (\cup_{l\in \{1,\dotsc,k\}\cap J_n}\sigma(x_{n,l})) \end{align} using the fact that $\cup_{l\in\{1,\dotsc,k\}\setminus J_n}\sigma(x_{n,l})$ is the trivial sigma algebra $\{\Omega,\emptyset\}$. Similarly, \begin{align} F_{k+j,n}^n=\sigma (\cup_{l=k+j}^n\sigma(x_{n,l}))\supseteq \sigma (\cup_{l\in\{k+j,\dotsc,n\}\cap J_n}\sigma(x_{n,l})) \end{align} for each $n,j$ and $k$. Therefore $\alpha'_n(j)\leq \alpha_n(j)\leq \sup_n \alpha_n(j)\to 0$ as $j\to\infty$, for every $n\geq n_0$.
Combining these results we conclude that $\alpha'_n(j)\leq \alpha_n(j)\leq \sup_n \alpha_n(j)\to 0$ as $j\to\infty$, for every $n\geq 1$.
Is it acceptable?