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Let us divide the (time) interval $[0,n]$ into $n/t$ subintervals of length $t$. Let us call the $k$th interval good, if, during that interval, the random walk spends time at least $t/5$ to the left o …

$S_n$ is a martingale with bounded jumps, and there is a result that it should either converge to a finite limit, or fluctuate, in the sense that $\limsup S_n=+\infty$, $\liminf S_n=-\infty$ (this, I …

All these questions are answered in paragraph 6 of Chapter III of Volume 1 of "An Introduction to Probability Theory and its Applications" by Feller. In particular: (1) $p=1/2$ is indeed the "right" …

It seems to me that this random walk is recurrent. Denote $Y_n=\|X_n\|$, where $(X_n, n\geq 0)$ is your "spiral" walk. Then, as $x\to \infty$, my calculations imply that $$ \mathbb{E}(Y_{n+1}-Y_n\mid …

Well, for $d\geq 2$, the projection of $S_n$ onto a hyperplane orthogonal to $\vec{p}$ is a zero-mean $(d-1)$-dimensional random walk with bounded jumps. Therefore, the answer to your question is ''ye …

No, it's not sufficient, you need assumptions on the tails of $\sigma$'s. See Chapter 5 of http://www.ime.unicamp.br/~popov/book_lyapunov.pdf

A couple of "one-dimensional" examples: https://arxiv.org/abs/1210.6328 and https://arxiv.org/abs/2209.00101