Let $(x_n)$ be a sequence in $\mathbb{R}$ and let $x*$ be its accumulation point. Then, one can find a subsequence $(x_{n_j})$ that converges to $x*$.
Now, let $(X_n)$ be a random sequence in $\mathbb{R}$. Do we have something like accumulation point for random sequences? If so, how would you define convergence for that? Is there a text book that convers my questions?