CLT and Slutsky for a mixed sum

Let $\{X_k\}_{k \ge 1}$ be a sequence of independent random variables such that $X_{2k-1} \sim \operatorname{Bin}(1, \theta)$ and $X_{2k} \sim \operatorname{Bin}(1, 1 - \theta)$, $k = 1, 2, 3, \dots$, where $\theta \in (0, 1)$. Let $\{Y_k\}_{k \ge 1}$ be another sequence of independent and identically distributed random variables such that $Y_k \sim \operatorname{Poisson}(\lambda)$, $\lambda > 0$. Define, for $n \in \mathbb{N}$, $S_{2n} = \sum_{k=1}^{n}(X_{2k-1} - X_{2k} + 1 - 2\theta)$, $W_n = \sum_{k=1}^{n} Y_k^2$ and $\sigma_{2n}^2 = 2n\theta(1 - \theta)$. Then which of the following statements is/are correct?