Consider the causal AR(1) time-series $y_t = \phi y_{t-1} + w_t$ such that $x_{t} - x_{t-1} = y_t$. I want to compute the limit $\lim_{t \to \infty} corr(x_t, x_{2t})$.
It is easy so see that the ACF of $y_t$ is $\rho(h) = corr(y_t, y_{t+h}) = \sigma_w^2 \phi^h\frac{1}{1-\phi^2}$. Therefore, for fixed $t$, translate $\{x_t\}$ such that $x_{t-1} = 0$ (we can do this because covariance is invariante unter adding constant). Thus $$corr(x_t, x_{2t}) = corr(y_t, \sum_{n =t}^{2t}y_t)=\frac{cov(\sum_{n =t}^{2t}y_t, y_t)}{cov(\sum_{n =t}^{2t}y_t, \sum_{n =t}^{2t}y_t)\gamma_y(0)} = \frac{\sum_{h=0}^t \gamma_y(h)}{\gamma_y(0)\sum_{s=0}^t\sum_{h=0}^s \gamma_y(h)}.$$
I am not sure how to continue from this. It seems like it is approximate to some double antiderivative of $\rho(h)$ over its antiderivative, which is like $\frac{\phi^{t+2}}{(t+2)\gamma_y(0)}$ which converges to $0$ for $t$ goes to infinity.
$$\begin{align} corr(x_t,x_{2t})&= \frac{\sum_{h=0}^t \gamma_y(h)}{\gamma_y(0)\sum_{s=0}^t\sum_{h=0}^s \gamma_y(h)}\\ &= \frac{\sum_{h=0}^t \phi^{h}}{\gamma_y(0)\sum_{s=0}^t\sum_{h=0}^s \phi^{h}}\\ &= \frac{\frac{1-\phi^{t+1}}{1-\phi}}{\gamma_y(0)\sum_{s=0}^t\frac{1-\phi^{s+1}}{1-\phi}}\\ &= \frac{1-\phi^{t+1}}{\gamma_y(0)\sum_{s=0}^t (1-\phi^{s+1})}\\ &= \frac{1-\phi^{t+1}}{\gamma_y(0)\left(t+1- \phi \frac{1-\phi^{t+1}}{1-\phi} \right)}\\ \end{align}$$
If $\phi<1$, the correlation converges to $0$.
If $\phi>1$, the correlation converges to $\frac{1}{\gamma_y(0)\frac{\phi}{1-\phi} }$.