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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 …

On a more elementary side, the are these probabilistic paradoxes, such as: https://en.wikipedia.org/wiki/Monty_Hall_problem, https://en.wikipedia.org/wiki/Two_envelopes_problem, https://en.wikipedia.o …

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 …

Let $S_n$ be the one-dimensional nearest neighbor random walk with $ 1-q=p=P[S_{n+1}=x+1\mid S_n=x]=1-P[S_{n+1}=x-1\mid S_n=x]$, where $p\neq q$. Then, there is a (rather surprising) fact that $Y_n=| …

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 - …

$S_n$ 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" …

Basically, the proof goes along the following lines: (1) Take a small $\varepsilon>0$ and show that the expected exit time from the interval $[-\varepsilon\sqrt{vl},\varepsilon\sqrt{vl}]$ is less tha …

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 …

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 …

Notice that the number of Catalan paths of area at least $cn^{\frac{3}{2}+\varepsilon}$ is less than the number of all paths that deviate from the horizontal axis by at least $n^{\frac{1}{2}+\varepsil …

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 …

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 …

Yes. Just observe that (1) on any fixed time interval the Brownian path intersects itself with positive probability (easy to see); (2) but the above implies that on any time interval the Brownian pa …

You can use the fact that $B_t-m_t$ is a reflected Brownian motion (see e.g. Revuz-Yor, Chapter VI, Theorem 2.3). I think it shouldn't be difficult to show that $$ \limsup_{t\to\infty} \frac{M_t-m_t}{ …