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

Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda …

Just a quick remark about another counterexample that one may construct: take a Simple Random Walk (on the integer lattice) in dimension $d\geq 3$ (so it is transient); then, an infinite set can be re …

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 …

The arrival and departure processes are obviously not independent: suppose that, with some very bad luck, no customers arrived to the system up to now; then (after the completion of the service of tho …

Let $\eta_t$ be the number of customers in the system at time $t$ and $\rho=\lambda/\mu<1$ be the load. It holds that $\eta_0+A_t-D_t = \eta_t$, so $A_t-D_t = \eta_t-\eta_0$. Write $$ A_t D_t = \frac{ …

Consider an $M/M/1$ queue with the arrival rate $\lambda>0$ and the service rate $\mu>\lambda$ (so that it is stable), in the stationary regime. Let $A_t$ be the number of arrivals in the time interva …

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 …

Let $A_r = \{z\in\mathbb{C}: |z|\leq 1\}\cup\{z\in\mathbb{C}: |z-1|\leq r\}$ be the unit disk with a small "bump" (I'm interested in the regime $r\to 0$). What can be said about the logarithmic capaci …

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 …

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 …

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 …

Doesn't it follow from the Lévy's continuity theorem? I mean, consider the characteristic functions of $S_{n-1}/s_n$ and $X_n/s_n$, the product of them converges to $e^{-t^2/2}$, so the characteristic …

There is an example in the book "Essentials of Stochastic processes" of Durrett, which shows that, in general, $m$ need not be equal to $2n-2$ (it is Example 4.5 of the chapter devoted to Markov chain …