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I am interested in the symmetric random walk on $\mathbb{R}$ which increments have the discrete law $$\mu=\sum_{i=1}^q p_i (\delta_{\omega_i}+\delta_{-\omega_i})$$ where the $p_i$ sum to $1/2$ and the $\omega_i$ are real numbers. This problem came up in an unrelated problem, and I was able to obtain the estimates I wanted (basically estimating the probability that the sum is close to 0, see How often a random walk with irrational increments is close to 0?). It basically depends on how well are the ratios between $\omega_i's$ approximable by rational numbers.

I wanted to see if I could attach that to a related body of litterature, but I have not found something relevant. It seems random rotations with irrational increments on the torus have been studied, but mostly for covering problems, and I have not been able to link both questions. I realise there are random walks in general Lie groups, with non-homogeneous increment law, but I am not enough at ease in these fields to see if there could be a connection.

Does anyone think of a (abstract) general problem that has a direct link with my initial problem?

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  • $\begingroup$ The problem is highly sensitive to the rational dimension of the rational vector space spanned by the set of $w$. Taking the usual transform, we obtain the polynomial (with real exponents) $\sum (x^w + x^{-w})p_w$. If for example, the dimension of the rational vector space is $k$, then this can be rewritten as a polynomial in $k$ variables with integer exponents, and thus corresponds to a random walk on ${\bf R}^k$. [Your examples are symmetric, but that is not relevant to this reduction.) $\endgroup$ Commented Jul 18, 2020 at 9:47
  • $\begingroup$ I agree, I think I will from the start assume that the $\omega$ are well suited for this problem to avoid the issue of a dimension $<q$, I know for instance that for a.a. $\omega\in \mathbb R^q$, the dimension is $q$, and they are badly approximable in the sense that for some $c,\varepsilon>0$, $|\omega\cdot j|>c|j|^{-q-\varepsilon}$ for $j\in Z^q$ $\endgroup$ Commented Jul 18, 2020 at 10:52
  • $\begingroup$ This is related to "small ball probability", and the inverse offord-littlewood problem, see cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/1/1221/files/…, not sure if you are aware of these similar type of results. $\endgroup$
    – shurtados
    Commented Aug 12, 2021 at 21:07

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The Cramer-Esseen theorem (Theorem 2 in paragraph 42 of Gnedenko--Kolmogorov book) seems to give a very sharp asymptotics for the terms in your sum in the linked answer, at least for large $n$.

Now, for small $n$, the terms in your sum do not actually go to zero as $\epsilon\to 0$ at all (for instance, $P(|S_2|=0)=\frac{1}{4}$), so it is not quite clear what kind of an answer you are looking for...

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  • $\begingroup$ I am actually looking for a reference regarding random walks whose increments are somehow irrational, as for the CE theorem, the $o(1/\sqrt{n})$ is actually too large as it does not depend on $\varepsilon$. I agree there might be a residual in the sum considered in the other question, but if you want to think without residual you can only sum on odd terms. $\endgroup$ Commented Jul 18, 2020 at 5:02

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