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Consider a sequence of i.i.d. random variables $(X_n)_{n\geq 0}$, and set $S_N = \sum_{n=1}^N X_n$. For which $p> 0$ do we have that \begin{equation} \lim \inf \frac{|S_N|}{N^{1/p}} > 0 \text{ almost surely}? \end{equation}

In [1], important results are shown for the quantity $M_N := \max_{n\leq N} |S_n|$. In particular, $\lim \inf \frac{|M_N|}{N^{1/p}}$ is almost surely $\infty$ for $p> \delta$ and almost surely $0$ for $p< \delta$, where $\delta$ is an index that is explicitly defined from the law of the $X_n$. The index $\delta$ is typically $\alpha$ for symmetric-$\alpha$-stable random variables.

However, the case I am interested in is presented as "much more difficult" by William Pruitt. I would be curious by any partial results (stable laws, infinitely divisible laws).

[1] Pruitt, The Growth of Random Walks and Levy Processes, 1981, Annals of Probability https://projecteuclid.org/euclid.aop/1176994266

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  • $\begingroup$ $\liminf$ or $\limsup$? $\endgroup$
    – Mikael de la Salle
    Commented May 1, 2018 at 13:50
  • $\begingroup$ lim inf! The lim sup is treated in the paper of Pruitt (as an easy consequence of the study with $M_N$). $\endgroup$
    – Goulifet
    Commented May 1, 2018 at 14:14
  • $\begingroup$ For stable processes $S_t$ (at least in continuous time), I suppose self-similarity helps: $Z_t = t^{-1/p} S_t$ is a self-similar Markov process, and the question is whether it stays away from zero or not. This seems to be a well-studied problem. Kyprianou's three papers on "Deep factorisation of the stable process" seem to be a good entry point into related literature. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jun 16, 2018 at 9:58

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