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Just a side comment: if you pass from binary to trinary, you can still obtain the Lebesgue measure by choosing the digits $0$, $1$, $2$, with probabilities $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ (som …

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 …

The process $\|B(t)\|$ is called $n$-dimensional Bessel process (or Bessel process with parameter $\nu=\frac{n}{2}-1$). I think formula $\bf 4$.1.1.4 of Borodin-Salminen "Handbook of Brownian Motion - …

It's not that simple. See about polar/nonpolar points/sets e.g. in http://wiki.math.toronto.edu/TorontoMathWiki/index.php/Brownian_Motion_and_Harmonic_functions If I remember correctly, a set is not …

Consider the following simple example of a function $f: \mathbb{R}\to\mathbb{R}$ which is open and discontinuous at all points. If $x\in\mathbb{R}$ is represented as something.$x_1x_2x_3\dots$ in the …

As for (3) (recurrence in $d\leq 2$ and transience in $d\geq 3$ of simple random walk), there are "electric networks"-proofs of these facts. See the classical book of Doyle and Snell "Random walks and …

No, the CLT need not hold under these assumptions. Consider the following example: take $p=1/2$ for definiteness, and divide the (discrete) time into intervals $I_1=[1,2]$, $I_n=(2^{n-1}, 2^n]$, $n\ge …

Heuristically, this probability should behave as $O(\sqrt{L/n})$, I guess. Observe that each queue, when not empty, is a random walk with zero drift, that actually moves once every $O(L^{-1})$ instanc …

Consider the standard two-dimensional Brownian motion, and define $\tau(A)$ to be the hitting time of $A\subset \mathbb{R}^2$. Let $hm_A$ be the harmonic measure (from infinity) on $A$. Let $B(r)$ be …

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