Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $$ -\infty = \liminf_nX_n<\limsup_nX_n=\infty\quad \mathsf P\text{-a.s.}? $$

share|cite|improve this question
up vote 2 down vote accepted

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 $-n\log(\log(n))$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.