Suppose I have two iid sequences of random variables $\{X_t\}_{t\ge1}$ and $\{Y_t\}_{t\ge1}$ that have full support. That is, if $A\subseteq\mathbb{R}$ has positive Lebesgue measure, then $P(X\in A) >0$ and $P(Y\in A)>0$. Such an iid sequence hits with probability 1 any set of positive measure for infinitely many $t$. That means that with probability one $$ \liminf_{t\rightarrow\infty} |X_t| = 0\qquad\text{and}\qquad \limsup_{t\rightarrow\infty} |X_t| = \infty . $$ Question: Is it possible to find a dependence structure between $\{X_t\}_{t\ge1}$ and $\{Y_t\}_{t\ge1}$ such that $$ P\left(\liminf_{t\rightarrow\infty} |X_t| + |Y_t| = \infty\right) > 0, $$ that is, with positive probability there exists no $M>0$ such that $|X_t| + |Y_t|<M$ for infinitely many $t$.
I have been trying to find a dependence structure but failed in doing so. I now believe that it might be impossible, but can't prove it.
Thank you in advance!