 Solution

This is the simple symmetric random walk on the integers, starting at zero. It is recurrent: it returns to zero infinitely often almost surely, and in fact visits every integer infinitely often. To see the two-sided unboundedness explicitly, gambler's ruin gives, for positive integers $a,b$,Letting $a\to \infty$ shows that $b$ is hit with probability one; the analogous argument hits every negative integer. From any state the same argument gives return to zero with probability one. Repeated use of the Strong Markov property then gives infinitely many returns and visits. ThereforeIn particular $X_n$ has no limit almost surely, and $|X_n|$ does not tend to infinity because of the returns to zero. In contrast, the strong law of large numbers gives $X_n/n\to 0$ almost surely, while the central limit theorem gives $X_n/\sqrt{n}\stackrel{d}{\to }N(0,1)$. These distinguish oscillation of the path from scaling limits of its probability distribution.