Random walk with zero drift

Question

Let $X_{k+1} = X_k+\xi_k$ be a random walk on $\Bbb R$ starting from $0$ and such that $\mathsf E\xi_0 = 0$, $\mathsf{Var}[\xi_0]>0$. Is that true that $$ -\infty = \liminf_nX_n<\limsup_nX_n=\infty\quad \mathsf P\text{-a.s.}? $$

Answer 1

If the variance is finite, the law of the iterated logarithm tells you that for i.i.d random variables $\xi_k$ with mean $0$, the limsup of their sum grows like the square root of $n\log(\log(n))$. A similar argument will show that the liminf grows like the square root of $-n\log(\log(n))$.