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See formula (7.15) on p.218 of Mörters-Peres "Brownian motion" (it is better suited for the case $r\to 0$ than (7.14) of Theorem 7.45 there).

Let $\lambda:=1-\epsilon_1$, $\mu:=1-\epsilon_2$; also, denote $p_L:=\frac{\lambda}{L}(1-\frac{\mu}{L})$ and $q_L:=\frac{\mu}{L}(1-\frac{\lambda}{L})$. Consider a fixed queue (one of those $L$), then …

In this situation the Foster-Lyapunov criterion still works. Let the state of the system be $n$ when there are $n>0$ customers in the system and the server is working, and $0$ when the server is under …

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 …

$S_n$ 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 …

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 …

Let $S(n)$ be the discrete sphere of radius $n$ (i.e., the internal boundary of the Euclidean discrete ball $B(n)$) centered in the origin, and consider a simple random walk starting at some $x\in\mat …

We now have a proof of a weaker result (only Hölderness), see Proposition 1.3 of https://arxiv.org/abs/1606.05805 . The Lipschitzness is still beyond our reach...

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 …

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{ …

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 …

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 …

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 …

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 …

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 - …